Optimisation and A MASS REDUCING FLOW FOR REAL-VALUED FLAT CHAINS WITH APPLICATIONS TO TRANSPORT NETWORKS

. An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the diﬀerence between the demand and supply distributions, represented by weighted sums of point masses. To accommodate eﬃciencies of scale into the model, one uses a suitable M α norm for transportation cost for α ∈ (0 , 1]. One then ﬁnds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this paper, we construct a continuous ﬂow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this M α mass reducing ﬂow for real-valued ﬂat chains by ﬁnding a higher dimensional real chain whose slices dictate the ﬂow. Keeping the boundary ﬁxed, this ﬂow reduces the M α mass of the initial chain and is Lipschitz continuous under the ﬂat- α norm. To complete the paper, we apply this ﬂow to transportation networks, showing that the ﬂow indeed evolves branching transport networks to be more cost eﬃcient. overlapping in cost eﬃcient ways, making ramiﬁed structures preferable. This is due to the fact that in operations there are often eﬃciencies of scale—i.e., the cost per pound for transporting freight decreases as the amount of freight increases.


Introduction
Transport theory as in the Monge-Kantorovich problem (1781;1942) focuses on minimizing transportation cost in a way that depends only on optimally allocating or distributing goods [26,28]. Here, the cost of a transport plan is typically an integral of some convex function of distance, such as |x − y| p for 1 ≤ p < ∞. And so, the functional does not account for the actual path traveled by the goods; one can assume each good is transported along a straight line connecting its source to its destination.
While this model allows for easier analysis, it is not realistic for many practical problems. As noted by Xia [41], the cost of transportation in real applications is rarely completely determined by the distance between the supply and demand locations. Instead, the physical transport path influences total cost because "carpooling" encourages overlapping in cost efficient ways, making ramified structures preferable. This is due to the fact that in operations there are often efficiencies of scale-i.e., the cost per pound for transporting freight decreases as the amount of freight increases.
Keywords and phrases: Optimal Transport Networks, Mass Reducing Flows, Flat Chains

Preliminaries
Flat chains generalize the concept of oriented manifolds, possibly with boundary. Their definition originates from investigating the following problem called The Plateau Problem: given a closed, (k − 1)-dimensional boundary Γ ⊂ R m , find a k-dimensional surface S of least k-dimensional area among all surfaces with ∂S = Γ. This problem was inspired by the Belgian physicist Joseph Plateau's (1801-1883) experiments with soap films [32]. In fact, questions about existence and regularity for a k-dimensional Plateau solution became the key motivation for the modern development of the field of geometric measure theory. To show the existence of a minimal surface S with the desired boundary, one might initially try to implement the direct method of calculus of variations-taking a minimizing sequence of surfaces with boundary Γ and extracting a subsequence converging to the solution surface. This would require existence of finite area surfaces with boundary Γ, lower semicontinuity of area with respect to some topology, and compactness with respect to this topology. Additionally, one needs continuity of the boundary operator to ensure that the boundary stays fixed. However, Figure 1.1 demonstrates how compactness of surfaces in R 3 can go awry.
Hence, mathematicians had to develop new strategies to answer the Plateau problem. In the 1930's, Douglas and Rado independently solved the problem for 2-dimensional surfaces that can be parameterized by a single disk; in particular, they showed that every smooth Jordan curve bounds a mapped-in disc of least mapping area [11,33]. Since their work heavily relies on conformal parameterizations of surfaces, it limits the types of possible singularities in solutions. In 1960, Reifenberg solved a formulation of the Plateau problem, but subject to certain topological constraints [34]. A natural objection to these solutions is that one must eliminate conceivable solutions that form in actual soap film experiments because of a priori imposed assumptions on achievable singularities and topological complexities.
To circumvent many of these restrictions, Federer and Fleming offered a solution to the Plateau problem for arbitrary dimension and codimension by using their newly defined normal and integral currents [18]. Fleming gave another approach in 1966 using different generalized oriented manifolds, called flat chains [20]. While avoiding the a priori assumptions on singularities and topological complexities, both currents and flat chains The space of surfaces (e.g. mappings) may fail to be compact with respect to a considered topology. Depicted above are three surfaces from a sequence with the unit circle as boundary whose surface areas decrease towards π. However, since each surface gains more and more thin tentacles, the limit could include all of R 3 . See [29]. offer natural topologies with compactness properties, and thus, surmount the issue in implementing the direct method of calculus of variations.
For more information on the history of the Plateau problem and its contribution to the founding of geometric measure theory, see [2,29].
One may consider finding an optimal transportation network as solving an M α mass generalization of the 1-dimensional Plateau problem. Motivated by this, we create an M α mass reducing flow on real-valued flat chains. But first we define in this section flat chains and develop the preliminary material necessary for creating the flow. Sources that provide much of the background presented here include [12,17,20,29,[36][37][38]. In particular, we follow the approach of Fleming [20] and White [37,38,40] in which we define flat chains for real coefficients with norm |g| α for α ∈ [0, 1] as the completion with respect to a flat metric of polyhedral chains. Chodosh's lecture notes from Brian White's Topics in Geometric Measure Theory course from Spring 2012 [9] provide an excellent expository treatment of the topics below including an equivalent definition to our fill-α norm defined in this section.

Flat chains, M α mass, and the flat-α seminorm
Let K ⊂ R m be a compact set in m-dimensional Euclidean space and let k ∈ {0, 1, . . . , m}. Definition 1.1. Let P k (K) denote the group of polyhedral chains of dimension k with real coefficients and support in K. We may write any polyhedral chain in P k (K) as P = g i σ i where each g i ∈ R and {σ i } is a set of non-overlapping, oriented, k-dimensional simplices in K. For any α ∈ [0, 1], define the M α k mass of P to be where H k is the k-dimensional Hausdorff measure. We simply write M α (P ) when the dimension is understood from context. Definition 1.2. Define the flat-α norm of a polyhedral chain P ∈ P k (K) by or equivalently, where ∂ is the usual boundary operation on polyhedral chains. The flat-α distance between P 1 , P 2 ∈ P k (K) is then defined to be F α (P 1 − P 2 ).
Most properties are straightforward to prove that the flat-α norm is indeed a norm on P k (K); the only property not straightforward is that F α (P ) = 0 implies that P = 0, which will follow after results in Subsection 1.6. (See Remark 1.25.) Also, note that F α (∂P ) ≤ F α (P ) by definition and considering if P = S + ∂R, then ∂P = ∂S.
At first glance, it may seem more natural to consider the topology induced by the M α mass instead of creating the flat-α norm. However, the following example elucidates why the flat-α norm is preferable: it more accurately captures the concept of geometric closeness for chains. Example 1.3. Consider polyhedral chains {P i } and P each consisting of a directed edge of length 1 and weight 1. Suppose that each P i is parallel to P and at (Euclidean) distance 2 −i from P . Since M α (P i − P ) = 2 for every i ≥ 0, the sequence {P i } fails to converge to P in the topology induced by the M α norm. However, the sequence does converge to P in the flat-α norm since where Q is the two-dimensional chain with weight 1 as in Figure 1  Note that Fleming's argument in [20] is extended in White's paper [38] to prove this lemma for polyhedral chains with coefficients in a more generalized normed abelian group G; so the above lemma follows from White's results by considering the chain coefficients as elements of the group R with group norm |g| = |g| α as White notes as an example in [38].
Using the flat-α completion of P k (K), we may now define general flat chains. Definition 1.5. Denote by F k (K) the F α -completion of P k (K) and call its elements real-valued, k-dimensional flat chains. Consider a sequence {P i } ⊂ P k (K) converging to T in the flat-α norm and note that Thus, {∂P i } is a convergent sequence of (k − 1)-dimensional polyhedral chains and we say that its limit is a (k − 1)-dimensional flat chain called the boundary of T , denoted ∂T . Remark 1.6. As in [38], one can then define F k (R m ) as the union of F k (K) over all compact sets K ⊂ R m , resulting in the fact that every real flat chain in R m is compactly supported. However, we only need to consider chains defined on compact sets for the construction of the flow as well as the application to transport networks.
In order to preserve the lower semicontinuity property of mass, we define the M α mass of a real-valued flat chain as follows.
Definition 1.7. For any real flat chain T ∈ F k (K), define its M α k mass to be We write M α (T ) when the dimension is understood from context.
This definition indeed coincides with the one above when T is polyhedral due to the lower semicontinuity of the M α mass on P k (K) with respect to the flat-α norm for polyhedral chains. Definition 1.8. And in general, we may extend the flat-α norm to real-valued, k-dimensional flat chain T by defining For convenience, we define N α (T ) := M α k (T ) + M α k−1 (∂T ) for T ∈ F k (K); and let N α k (K) := {T ∈ F k (K) : N α (T ) < ∞}, the set of all real-valued, k-dimensional flat chains with both finite M α mass and finite M α boundary mass.
Once again, most properties are straightforward to verify to prove that this extended definition of the flat-α norm is indeed a norm on F k (K) (except for that discussed in Remark 1.25). We also still have the property that F α (∂T ) ≤ F α (T ). This next lemma on lower semicontinuity follows from the definition of the M α mass, considering again the chain coefficients as elements of the group R with group norm |g| = |g| α and shown in [20,38]. Lemma 1.9. [20,38] On F k (K), the M α mass is lower semicontinuous with respect to the flat-α norm.
Comparing the flat-α norm and the norm induced by the M α mass on real-valued flat chains, one finds that convergence in the M α norm is stronger than that in the flat-α norm when k < n since F α (T ) ≤ M α (T ). Example 1.3 provides an instance where convergence in the flat-α norm does not imply convergence in the M α norm. Thus, we will use the flat-α norm to measure closeness of two chains since it takes into consideration the M α mass of the chains as well as their geometric nearness in K ⊂ R m .
An additional convenient property of the definitions of flat chains is the following lemma, which says that there exist polyhedral chains P i converging to T in the flat-α norm such that M α (T ) = lim i→∞ M α (P i ). The proof of this lemma follows from the definition of the M α mass of a flat chain as well as results from Subsection 1.

Compactness
Recall that Fleming defined flat chains and the flat-α norm in order to have the necessary compactness property for employing the direct method of calculus of variations. Using a deformation theorem shown in [37], White proved a more generalized compactness theorem for chains with coefficients in abelian groups with certain properties in the appendix of [40], which we provide for convenience below: Theorem 1.11. [40] The set of flat chains of finite generalized mass and boundary mass with coefficients in an abelian group G is compact with respect to the generalized flat norm if and only if G has the property that balls are compact in G; i.e. {g ∈ G : |g| ≤ r} is compact for all r > 0.
The following corollary immediately follows as it is merely a rephrasing of this theorem to fit our context of real-valued flat chains. That is, we consider G = R with metric |g| α for α ∈ (0, 1]. (Note that balls are indeed compact in R with this metric.) Then there exists a subsequence of {T j } converging with respect to the flat-α norm to a real flat chain T ∈ F k (K). Remark 1.13. Compactness fails in the case α = 0 since {g ∈ R : |g| 0 ≤ r} = R for all r ≥ 1. Since we will employ the direct method of calculus of variations at several points in the construction of the flow, we will often refer to Corollary 1.12, and consequently, the M α reducing flow will be restricted to α ∈ (0, 1].

Cartesian products of flat chains
Note that most of this material on Cartesian products can be found in Fleming's paper developing the theory of flat chains over a finite coefficient group [20], but we include for convenience this development of Cartesian products of real flat chains (with coefficient group with norm |g| α for α ∈ [0, 1]).
In this paper, we will need the Cartesian product of flat chains to create a sequence of ("birthday cake shaped") higher dimensional chains in R + × R m within the construction of the flow (in Subsection 2.2). While we cannot give a satisfactory definition for the Cartesian product of two arbitrary flat chains, we may provide one for when at least one of the chains is in N α k (K), which is all we need for the construction of the flow. To do so, we must first define the Cartesian product of two polyhedral chains and then give a bound on the flat-α norm of their Cartesian product. Definition 1.14. Suppose P is a k-dimensional polyhedral chain in K 1 ⊂ R m and Q is an -dimensional polyhedral chain in K 2 ⊂ R n . Their Cartesian product, denoted P × Q, is a (k + )-dimensional polyhedral chain in since for α ∈ [0, 1], |g i g j | α = |g i | α |g j | α for {g i } and {g j } the sets of real coefficients for the polyhedral chains P and Q, respectively. Moreover, To define the Cartesian product in more generality, we will need the following lemma and Lemma 1.10. . For P and Q as above, F α (P ×Q) ≤ N α (P )F α (Q) and likewise, F α (P ×Q) ≤ F α (P )N α (Q).
Proof. Here, we will replicate the proof found in Fleming's paper [20] for the reader's convenience. Suppose Q = U + ∂V for U ∈ P (K 2 ) and V ∈ P +1 (K 2 ). We wish to consider the flat-α norm of P × Q = P × (U + ∂V ). First, we will show that F α (P × ∂V ) ≤ N α (P )M α (V ). Note that by Equation 1.2, Thus, by definition of the flat-α norm, we have So, we have, again using Equation 1.1, where the last inequality follows from the fact that M α (P ) ≤ M α (P ) + M α (∂P ) = N α (P ).
Since we showed the above for all U ∈ P (K 2 ) and V ∈ P +1 (K 2 ) such that Q = U + ∂V , we must have F α (P × Q) ≤ N α (P )F α (Q), as desired. Definition 1.16. Now we may define the Cartesian product of T ∈ N α k (K 1 ) for compact set K 1 ⊂ R m and S ∈ F (K 2 ) for compact set K 2 ⊂ R n . We first define T × Q for any Q ∈ P (K 2 ) as follows. Let {P j } ⊂ P k (K 1 ) converge to T in the flat-α norm on F k (K 1 ) so that N α (P j ) → N α (T ). (Lemma 1.10 shows that there exists such a sequence.) Then by Lemma 1.15, we have for any Q ∈ P (K 2 ) So, {P j × Q} ⊂ P k+ (K 1 × K 2 ) converges in the flat-α norm on F k+ (K 1 × K 2 ) to the real-valued, k + dimensional flat chain T × Q. For our purposes, this definition of a Cartesian product of a flat chain and a polygonal chain is sufficient; however, we include the following discussion to define the Cartesian product in more generality simply for completeness of exposition. Now take {Q i } ⊂ P (K 2 ) converging to S ∈ F (K 2 ) in the flat-α norm on F (K 2 ). Since we have N α (P j ) → N α (T ), we can extend the result of Lemma 1.15 to real-valued, k-dimensional flat chain T in place of the polyhedral chain P . Note that then We have similar properties for this more generalized definition of Cartesian product of flat chains as we had for polygonal ones. Let T ∈ N α k (K 1 ) for compact set K 1 ⊂ R m and S ∈ F (K 2 ) for compact set K 2 ⊂ R n . We have the operation × is bilinear and which can be seen in the example of Figure 1 Recall that within the construction of the flow (in Subsection 2.2), we will need the Cartesian product to create a sequence of ("birthday cake shaped") higher dimensional chains in R + × R m . (See Figure 2.1.) So, we are particularly interested in the case with S ∈ F (K 2 ) for compact set K 2 ⊂ R n as before and for r < t, denotes a positively oriented, polygonal chain of appropriate dimension with weight 1. Thus, T = [|r, t|] is the interval (r, t) considered as a positively oriented, polygonal chain of weight 1. Because the flow construction relies on this special case, we will explicitly explore the above discussion in this context. Here, [|r, t|] × S ∈ F 1+ (K 1 × K 2 ) and its boundary is which also leads to Example 1.17. The above equations can be further exemplified using Figure 1

Push forward of real flat chains by Lipschitz maps
Since the construction of the flow will require a "birthday cake" decomposition, we will need finite M α mass cycles to bound higher dimensional chains achieving the considered norm. For integral chains (i.e., chains with integer coefficients), this fact would hold for the flat-α norm because of the isoperimetric inequality [8]. However, real-valued chains do not possess an isoperimetric property. So, we will define a new norm, called the fill-α norm, using cones over chains. Then, we will show that finite cycles bound higher dimensional chains achieving this new norm. In order to define cones over chains, we will need to first define push forwards of real-valued chains and give the homotopy formula, which we do now. Moreover, we will need push forwards of chains under projection maps, so we will be particularly interested in Lipschitz maps in the following sections.
If g : R m → R n is a piecewise linear map, then it induces a chain homomorphism g : P * (K 1 ) → P * (K 2 ) for compact sets K 1 ⊂ R m and K 2 ⊂ R n , and where * represents any chain dimension as in [38]. This map is defined by g : so that g respects the chain grading, g (S + T ) = g S + g T for S, T ∈ P * (K 1 ), and g commutes with the boundary map ∂. For simplified notation, we write g T to represent the image of T under g . This map on polyhedral chains may be extended uniquely to real flat chains if g is Lipschitz with respect to the M α norm: Then, g is also Lipschitz with respect to the flat-α norm consequently inducing a unique extension to Proof. Let T ∈ F k (K 1 ). Then since g is Lipschitz with respect the the M α norm, we have for R ∈ F k+1 (K 1 ), And so, the desired inequality, Similarly, we may extend a Lipschitz map f : ) by approximating f by piecewise linear maps, resulting in a similar lemma. The proof of this lemma follows a similar argument as that found in [20]. and Let K ⊂ R m be compact and let f, g : K ⊂ R m → R m be smooth mappings. The affine homotopy of f to g is the map h : Rearranging the above equation gives the homotopy formula: (1.7)

Cones over flat chains
Now that we have defined push forwards of real-valued chains and given the homotopy formula, we can define cones over chains. Essentially, the construction of the flow requires finite M α mass cycles to bound higher dimensional chains achieving the considered norm, and for integral chains this is shown by an isoperimetric inequality, which is not true for real-valued chains. So, we are working towards Corollary 1.22 in order to define the new needed norm defined in the next section. Fix x 0 ∈ K for K ⊂ R m convex, and thus, star-shaped with respect to the point x 0 (i.e., tx + (1 − t)x 0 ∈ K for each x ∈ K and each 0 ≤ t ≤ 1.) Then, the cone over T , denoted cone(T ), is given by h ([|(0, 1)|] × T ).
It directly follows that cone(T ) ∈ F k+1 (K), and by the homotopy formula, (1.8) Because of this, every cycle bounds its cone, and the following lemma implies that this chain will also have finite M α mass if the cycle does.
Proof. This follows from by Equation 1.5. In [37], White proves a similar result as the above Corollary to similarly extend Almgren's isoperimetric inequality [1] and Fleming's isoperimetric inequality [20].

Defining a filling norm for cycles
Recall that we need finite M α mass cycles to bound higher dimensional chains achieving some norm since the construction of the flow requires a "birthday cake" decomposition. Using the above results, we can define this new norm, which we will call the fill-α norm, and show that finite cycles indeed bound higher dimensional chains achieving this norm. While this is a new norm in a sense, it is merely a simplified flat norm for cycles; in fact, the fill-α norm and flat-α norm are equivalent for cycles, and thus, either may be used to capture the idea of "closeness" of two chains with the same boundary. The only instance of such a norm (defined for polyhedral chains) that the author is aware of is in the lecture notes by Chodosh on Brian White's Topics in Geometric Measure Theory course from Spring 2012 [9]. However, this section's results, while necessary for the construction of the flow, do not provide novel insight into real flat chains since the new norm is equivalent to the flat norm on chains.
As before, assume K ⊂ R m is convex, and for convenience, denote the cycles in F k (K) by Z k (K); i.e., By the above definition of cone, we know the following norm is well-defined.
Definition 1.23. For T ∈ Z k (K) with compact support, define a filling-α mass for T as The following lemma shows that the fill-α norm and the flat-α norm are equivalent for cycles. The proof follows the same argument found in [9].
where c is a constant depending only on K ⊂ R m .
Proof. The first inequality follows from the fact that the infimum on the right-hand side is taken over a subset of (k + 1)-dimensional flat chains while the infimum on the left-hand side is taken over all (k + 1)-dimensional flat chains. For the second inequality, write T = Q + ∂R. Note that ∂T = ∂Q = 0, and so, Q = ∂(cone(Q)) by Equation 1.8. Thus, T = ∂(cone(Q) + R). By the triangle inequality and Lemma 1.21, Taking the infimum over all such Q, R proves the second inequality for constant c := max{diam(K), 1}.
Remark 1.25. This lemma may be used to complete the argument that the flat-α norm is indeed a norm as it may be used to show that F α (T ) = 0 implies T = 0. Additionally, it is needed for the proof of Lemma 1.10.
Another consequence of this lemma is that the fill-α norm is lower semicontinuous with respect to the flat-α norm: , the Fill α mass is lower semicontinuous with respect to the flat-α norm.
We will need in the flow construction to apply a projection, so we will need a bound on the fill-α norm under a Lipschitz map in addition to Lemma 1.19. That is, we need the following lemma on how a Lipschitz map f : R m → R n extends to a map on flat cycles that is Lipschitz with respect to the fill-α norm. This lemma follows from the fact that cycles map to cycles (∂ and f commute [19]) and Lemma 1.19.
Note that there is a slight complication above in ensuring the fill-α norm on Z k (f (K)) is well-defined on the left hand side of the inequality since f (K) may not be convex; but, you may consider the convex hull without any loss of generality or consider Z k (R n ) using Remark 1.6.
Keep in mind that when defining the discrete sequences in the flow, we will use the fill-α norm to capture the idea of "closeness" of two chains with the same boundary (in functional G that will be defined in Equation 2.1). Moreover, for the "birthday cake" decomposition, we will need the existence of a higher dimensional chain achieving the fill-α norm between the two sequential chains forming its boundary, and thus, need the following theorem. Notice that this is restricted to α ∈ (0, 1] because its proof relies on the Compactness Theorem (Theorem 1.12). First, we state a necessary lemma for the proof. Proof. From Corollary 1.22, we know Fill α (T ) = inf{M α (Q) : ∂Q = T with sptQ ∈ K} < ∞, and so, we may take a minimizing sequence

Slicing of real flat chains
The M α mass reducing flow will be defined as a real flat chain of finite mass whose slices dictate the flow. Hence, we must define and give some properties for slicing real-valued flat chains. Of particular importance to the M α mass reducing flow will be Lemma 1.31. For further background in slicing, see De Pauw and Hardt's paper on flat chains in metric spaces [12] and White's paper [38]. Definition 1.29. Let T ∈ N α k (K), t ∈ R, and f : K ⊂ R m → R be Lipschitz. Then we may define the slice of T as where represents restriction.
Lemma 1.30. [38] Suppose that α ∈ (0, 1], T ∈ N α k (K) and p : R m → R is an orthogonal projection. Then, for α ∈ (0, 1], and suppose that the sequence {T j } converges to T in the flat-α norm. Let p : R m → R be an orthogonal projection. Then, for almost every t ∈ R, there is a subsequence j such that Hence, lim j →∞ F α ( T j , p, t + − T, p, t + ) = 0; i.e., the slices of the T j converge in the flat-α norm to the slice of T .
Proof. The main pieces of the argument presented in [12] include first passing to a subsequence to ensure that ∞ j =1 F α (T j − T ) < ∞, and then, the result then follows from the fact that under these conditions and Lemma 1.30.
Finally, the following corollary results from the above lemma and lower semicontinuity of the M α mass (Lemma 1.9). Corollary 1.32. [12] Under the same assumptions in Lemma 1.31, we have for almost every t ∈ R.

The Flow
In this section, we construct an M α mass reducing flow for real-valued flat chains. That is, the entirety of Section 2 constitutes the proof of the main theorem of this paper, Theorem 2.1, by giving the construction of this flow. For the rest of the paper, we will always assume that α ∈ (0, 1] and that K ⊂ R m is compact and convex. Theorem 2.1. For any fixed C ∈ R + and any T 0 ∈ N α k (K), a real-valued flat chain with compact support contained in some compact, convex set K ⊂ R m , there exists a flow giving real-valued, k-dimensional flat chains T t for almost every t ∈ [0, C] so that ∂T t = ∂T 0 , and for almost every 0 ≤ t < r ≤ C.
The above T t will be slices of a higher dimensional chain found by taking a limit of "birthday cake" decompositions created out of discrete sequences of real-valued flat chains. In each sequence, each chain minimizes a functional that tends to reduce M α mass while ensuring that each chain stays geometrically close to the previous one in the sequence. Note that this flow is interesting for chains T 0 not a local minimum to this functional, as we'll see in examples.
While the proof of this theorem is this entire section of the paper, we briefly reiterate the outline of the construction here in more detail for convenience of the reader. First, we will find step minimizing sequences of chains in Subsection 2.1, and out of these sequences, we will create "birthday cake" decompositions, S h 's, at the beginning of Subsection 2.2. Then, we will take a limit of these higher dimensional chains to find real flat chain S in Theorem 2.6. The "time slices" T t of S are the chains mentioned above in Theorem 2.1. Theorems 2.7 and 2.8 will show that the flow dictated by these T t indeed possess both the mass decreasing property and the Lipschitz continuous with respect to the flat-α norm property stated in Theorem 2.1.
A helpful analogy for intuition behind this construction is children's flip books and animations. Essentially, the discrete minimizing sequence of chains can be considered as pages of a flip book, while the higher dimensional chains S h can be conceptualized as sewing these pages together into a flip book. When flipping through this flip book, one would see a sequence of chains depicted moving from an original chain to a locally optimal one in such a way that preserves boundary. The smaller the h, the "smoother" this movement would appear when flipping through the book because the drawing on the subsequent page will only be a slight variation of the previous. Then, S would represent the limit of these flip books (taking h → 0) resulting in a "smooth" looking animation having time stamp depicting T t at time t. The result is a "nice" animation flowing from the original chain to an M α locally optimal one in such a way that preserves the boundary of the original.

Step minimizing sequence
We begin by creating the discrete minimizing sequence of chains for each fixed step size. Recall that we will always assume that α ∈ (0, 1] and that K ⊂ R m is compact and convex. We will begin with a fixed step size h > 0 and let T 0 ∈ N α k (K) have compact support. We will then inductively define a sequence of real-valued flat chains {T h j } so that T h 0 = T 0 and T h j (step with respect to h) minimizes a functional that reduces the M α mass of a chain; however, we will require that this sequence keeps the boundary fixed as well as ensuring that the next chain is geometrically close to the previous, by including a term involving the fill-α mass in the functional. To do this, we will need to define the following functional which has two terms: one involving the M α mass and the other involving the fill-α mass. The following theorem states that there exists a minimizer to this functional, which we will then apply to each H = T h j to construct our discrete minimizing sequence of step size h.
Given H ∈ N α k (K) with compact support in K, define which is non-empty since H ∈ C. Set L := inf{G H (T ) : T ∈ C}, which is finite because H ∈ C. Also let {R i } ⊂ C be a minimizing sequence such that Because of this and since {R i } ⊂ C, we have Thus, by the compactness Corollary 1.12, there exists a subsequence {R i } which converges in the flat-α distance to some real flat chain, say R. By the definition of infimum and since the M α and Fill α masses are lower semicontinuous with respect to the flat-α norm, we find We also have where the limit is with respect to the flat-α norm. Now, using R, define R so that its support is contained in K. Let f : R m → K be the nearest point projection map so that |f (x) − x| = inf{|y − x| : y ∈ K}.
Note that f is a Lipschitz map with Lipschitz constant no greater than 1. Set R := f R. By Lemma 1.19, Lemma 1.27, and the property that f (x) = x for every x ∈ K, we know Moreover, Proof. This follows from defining R in such a way that ensures We now inductively define the sequence of real-valued flat chains {T h j } for the construction of the flow. Consider again T 0 ∈ N α k (K) having compact support. First, set H = T h 0 = T 0 ; and then, set T h 1 := arg min{G H (T ) : T ∈ N α k (K) with compact support in K, ∂T = ∂T 0 }, which we know exists by Theorem 2.2. Continue to inductively apply Theorem 2.2 to each H = T h j−1 to find T h j . By doing so, we have the following properties: Lemma 2.4. The mass of the step minimizing sequence is non-increasing; i.e.

Additionally, we have the estimate
Proof. This lemma directly follows from the construction defined above. First Corollary 2.

Construction of the flow
To finish the construction of the flow, we will find a higher dimensional chain whose slices will dictate how to reduce the M α mass of the initial chain. Recall that we will do this by first creating the "birthday cake" decompositions S h out of each step minimizing sequence constructed above. The limit of these decompositions will result in our desired higher dimensional chain S whose slices dictate the flow.
For every step h > 0, we know from Lemma 2.
by the estimate found in Lemma 2.4. Using a "birthday cake" decomposition, construct a (k + 1)-dimensional, real-valued chain in R + × R m as follows. Let  Computing the boundary of S h with analogous techniques of Section 1.3, we find Ultimately, we wish to take a limit of these S h as h → 0, but in order for such a limit to converge, we must restrict S h to some finite time interval cross R m so that the sequence of these higher dimensional chains has a bounded supremum of M α mass and M α boundary mass.
Lemma 2.5. Given C ∈ R + and fixed h > 0, then the M α mass and M α boundary mass of the restriction S h [0, C] × R m is bounded and Proof. Given C > 0 and h > 0, choose integer so that ( − 1) √ h < C ≤ √ h. Using the triangle inequality, Equation 1.5, Lemma 2.4, and Inequality 2.2, compute Computing the boundary of this restricted chain, one can show that So, using Equation 1.5 and Lemma 2.4, we find Combining these bounds, we find Using this lemma, we will now find our limiting chain S whose slices will dictate the flow.
By the Compactness Corollary 1.12, there exists a subsequence {S i } converging to some real flat chain S in the flat-α norm.
Since S i converges to S in the flat-α norm, ∂S i → ∂S in the flat-α norm. In the proof of Lemma 2.5, we computed for each h = 2 −i , where ∈ Z depends on h and ( − 1) Since {∂S i } converges with respect to the flat-α norm, we have {T h } for h = 2 −i converge with respect to the flat-α norm to some real flat chain T C ∈ N α k (K) with ∂T C = ∂T 0 . See Figure 2.2 for a visual representation of S. Now, we will define the slices of S that will dictate the flow and show that these slices have the desired properties: mass decreasing, boundary preserving, and Lipschitz continuous with respect to the flat-α norm.
Let p : R + × R m → R + be the projection defined by p(t, x) = t. Denote the slice of S by p at t > 0 by S, p, t + . Note that this slice is a finite M α mass, k-dimensional flat chain for almost every t > 0. Similarly, we may define the slices of S i for each i and again have S i , p, t + ∈ N α k ([|0, C|] × K). See Figure 2.3 for a representation of these slices.

Set
T t := π S, p, t + where π : R × R m → R m is the projection map defined by π(t, x) = x. We will utilize the slicing results, Lemma 1.31 and Corollary 1.32, to prove the following theorems: Theorem 2.7. Given C ∈ R + . For almost every t such that 0 ≤ t ≤ C, and Proof. Note that Lemma 2.4 already shows this result for when t is a dyadic rational number. Otherwise, by Figure 2.3.) By the Slicing Lemma 1.31, we know that for almost every t > 0, there exists a subsequence i in which the convergence is "quick," that is, Thus, for all such t that are not dyadic rationals, we have for h = 2 −i F α (π ( S i , p, t + ) − π ( S, p, t + )) → 0; By the lower semicontinuity of M α mass of slices with respect to the flat-α distance (Corollary 1.32) and by Lemma 2.4, Moreover, by construction, we have for almost every t ∈ [0, C] where h = 2 −i and the limit is with respect to the flat-α norm.
Of course, the interesting examples are when the inequality in Theorem 2.7 is strict, which happens when T 0 is not a local minimum to the functional G defined in Equation 2.1. We will explore multiple such non-trivial examples in which mass actually decreases in the following section. But first, we will complete the proof of Theorem 2.1 by proving the remaining Lipschitz continuous property: Theorem 2.8. Let C ∈ R + . The constructed flow is Lipschitz continuous in time with respect to the flat-α norm; i.e., for 0 ≤ t < r ≤ C, Thus, Since the flat-α norm is lower semicontinuous, then for {i } as in the proof for Theorem 2.7, we have To see the last inequality, choose ,˜ so that where this last equality follows from the definition of projection π and the fact that H k+1 (T h j ) = 0 (since each T h j is k-dimensional). Finally, using the triangle inequality, Inequality (2.2), and our choice of ,˜ , we find that the above is bounded by finishing the proof.
This concludes the proof of the paper's main Theorem 2.1, but we remind the reader here of its outline as well as the intuitive analogy of flip books. Recall that in Subsection 2.1, we first found step minimizing sequences of chains, which we conceptualized as the pages of flip books. Out of these sequences, we created "birthday cake" decompositions, S h 's, at the beginning of Subsection 2.2, and we conceptualized these as sewing together those pages into flip books. Finally, we found real flat chain S in Theorem 2.6 through a convergence argument, and we imagined S to be the animation obtained from the limit of the flip books as each page changes less and less from the previous page (h → 0). The "time slices" T t of S are the chains in Theorem 2.1, and Theorems 2.7 and 2.8 showed that the flow dictated by these T t indeed possess the properties claimed in Theorem 2.1: the flow is mass decreasing, boundary preserving, and Lipschitz continuous with respect to the flat-α norm.
For transport networks, as explored in Section 3, the "flip book-like" constructed flow would appear to reduce M α cost through a "(un)zipping" process while not altering the boundary. For example, time stamps of this process on a particular transport path may be seen in Figure 0.1; in a sense, these time stamps could be imagined to be pages of such a flip book animation described above.

Applications to Transportation Networks
We wish to investigate the structure of optimal branched transport paths that accommodate efficiencies of scale into the model, and thus, we will consider such paths as real-valued, 1-dimensional flat chains with an M α cost function. Instead of solving this optimization problem outright, we will apply our constructed cost reducing flow to evolve transport networks. We will see in examples how this minimizing flow encourages overlapping structures because of the M α mass term in functional G in Equation 2.1. Thus, the functional G is particularly amenable to evolving branched transport networks in cost reducing ways through a "(un)-zipping" process, creating or eliminating interior vertices. In fact, the ability of our geometric flow to alter the network's underlying topology is particularly advantageous since many of the computational complications of current numerical methods stem from not a priori knowing the topology of the optimal network. It is our hope that the constructed flow, especially the discrete step minimizing process, helps further research on computational methods of optimizing branched transport networks.
Since the underlying theme of geometric flows is to gain insight into difficult optimization problems through indirect methods, minimizing flows are only able to find local optimal solutions. Essentially, minimizing flows are only able to describe how to reduce the functional of the currently considered object and are unable to foresee if first increasing and then decreasing that functional would actually lead to a better solution. Our constructed flow is no different and will not guarantee convergence to the optimal solution but instead evolve an initial network to a locally optimal branched network. However, this "nearest" locally optimal solution might be significantly geometrically and topologically "far away" from the initial path.
In this section, we will first give background on the branched transport problem and then analyze a few examples showing how the flow evolves networks. The example developed in Subsection 3.2.1 shows an initial path flowing to the global optimal solution. In Subsection 3.2.2, we will see an example merely flowing to a local minimum to show that indeed our minimizing flow is no different than others in this regard. However, we will also see in that subsection how certain shapes of initial transport networks appear to always flow to the global optimal solution even in strikingly different boundary configurations.

Preliminaries for transportation networks
Let µ + , µ − be two atomic measures of equal mass supported in closed and convex K ⊂ R m ; say are called sources and the {v j } j=1 are called sinks.
Definition 3.1. A transportation network, or transport path, from µ + to µ − is a weighted directed graph P whose vertex set V (P ), directed edge set E(P ), and weight function λ : E(P ) → (0, +∞) satisfy {w i } k i=1 ∪ {v j } j=1 ⊂ V (P ) and for any vertex x ∈ V (P ), where e − and e + denote the starting and ending endpoints of edge e ∈ E(P ), respectively.
Let P ath(a, b) denote the space of all transport paths from µ + to µ − , which is always nonempty because the following "coned" network satisfies the above conditions: where x ∈ K. Note that one may consider a transport path P from µ + to µ − as a 1-dimensional, real-valued polyhedral chain as in Definition 1.1. Thus, P ath(a, b) ⊂ P 1 (K). The balance equation (3.1) can be thought of as Kirchhoff's law or as mass being conserved at each vertex. In terms of polyhedral chains, this means that ∂P = µ − − µ + when one views µ + , µ − as 0-dimensional chains.  For continuous extensions of Gilbert's model, see [5,27,41]. However, since our flow requires the initial chain to have finite boundary mass, we only focus on transportation networks between finite atomic measures. Additionally, we may restrict our attention in this section to transport paths as polyhedral chains because we may approximate any flat chain by a polyhedral chain in the flat-α norm. In fact, we will further restrict our attention to only polyhedral chains without cycles. Note that the constructed flow does not guarantee that every chain T t from the flow has a tree structure, and it may indeed be advantageous in functional G in Equation 2.1 to sacrifice an increase in the M α mass term to carve out some area using a cycle to decrease the fill-α term. To simplify investigating the flow on paths using analytic methods, we will restrict our attention to polyhedral chains without cycles resulting still in flowing the initial chain to local minima due to Lemma 3.3.
Note that since the flow required α ∈ (0, 1] for the convergence arguments to hold, we will only be able to investigate reducing the M α cost for these α values. Thus, our flow will not apply to finding solutions to the case where α = 0, famously known as Steiner trees.

Examples
When applying the flow to some initial transportation network, one transforms the network, possibly changing the topology of the initial path through a "(un)zipping" process, creating or eliminating bifurcation points in cost reducing ways. As discussed above, we will restrict our attention here to only polyhedral chain transport paths without cycles. We will analyze examples where the supply and demand distributions are two sources/one sink and two sources/two sinks, as the optimal paths are known in these cases.

Two sources; one sink
Let α ∈ [0, 1], µ + = m 1 δ w1 + m 2 δ w2 , and µ − = (m 1 + m 2 )δ v , where m 1 , m 2 ∈ R and w 1 , w 2 , v ∈ R 2 are distinct. In this case, we know the optimal transportation network has either a "V" shape, an "L" shape, or a "Y" shape [6,41]; for the "Y" shaped case, we further know the location of the optimal branching point as given in the lemma below.
Lemma 3.4. [41] If P ∈ P ath(µ + , µ − ) is optimal and made up of three edges, then the interior vertex B of P must satisfy the following angle constraints: and

4)
where k 1 = m1 m1+m2 , k 2 = m2 m1+m2 , and θ 1 , θ 2 are as shown in Figure 3.1. We will examine how the flow evolves an initial path having a "V" shape to the optimal transport path. Again, Let µ + = m 1 δ w1 + m 2 δ w2 and µ − = (m 1 + m 2 )δ v , where m 1 , m 2 ∈ R and w 1 , w 2 , v ∈ R 2 are distinct, but this time let α ∈ (0, 1]. Let's additionally assume that the optimal transport network has a "Y" shape. Without loss of generality, set v = (0, 0) and let the positive x-axis extend from v to B = (x B , 0) where B is the optimal bifurcation point for the "Y" shape; that is, the x-axis breaks the angle ∠w 1 Bw 2 into θ 1 and θ 2 in Lemma 3.1.
Let us see how the flow zips up the initial path T 0 ∈ P ath(µ + , µ − ) having the "V" structure; i.e., Lemma 3.5. One may choose each T h j for every h > 0 in the discrete step minimizing process to be a "Y" shaped path with bifurcation point on the x-axis, and thus, ensure that each T t for almost every t > 0 has this same "Y" shaped structure.
Proof. We may assume that each path in the discrete sequences is contained in the convex hull of w 1 , w 2 , and v [5]. Now fix h > 0. We will prove via induction that every polyhedral path in our discrete sequence is "Y" shaped and its interior vertex lies on the x-axis. Let where B j−1 = (x j−1 , 0). (For j = 1, B j−1 = (0, 0) so that T h j−1 is exactly T 0 .) Suppose T minimizes functional G in Equation 2.1 (for H = T h j−1 ), and let A = (x, y) be its interior vertex. (For the existence of such an A, we are relying on our restriction to polyhedral chains without cycles.) By Lemma 1.19, we may use projection to decrease the M α mass (and also, the Fill α norm), and so, we may assume that A lies in the convex hull of v, w 1 and w 2 . Without loss of generality, assume that the edge [|A, v|] intersects [|w 1 , B j−1 |] at A = (x,ỹ). See Writing the above equation in terms of (x, y) (including expressing (x,ỹ) in terms of (x, y)), one can show that the above function is convex in y. And so, we have Hence, there is a minimizer of G that is "Y" shaped and its interior vertex lies on the x-axis, and we may choose this minimizer as T h j in the flow's construction. Moreover, the flow will stop zipping once the path has reached the optimal shape. One may see this by examining how any perturbation of the network with the optimal bifurcation point will increase the functional in Equation (2.1).
Remark 3.6. In fact, using that perturbation argument, one could show that if the initial "V" shaped network is the optimal one, the flow will keep this network fixed. And similar to the argument in Lemma 3.5, one may show that the flow unzips initial "Y" shaped paths with angles greater than that in Lemma 3.4 until the optimal bifurcation point is reached (or completely eliminating the bifurcation point if the "V" network is optimal). Furthermore, one may show that the flow zips up a "V" shaped initial path to the "L" shaped optimal network using the same technique. In this case, one would see a creation and subsequent elimination of an interior vertex in the networks as they flow from the "V" shape to the "L" shape. Thus, the flow always evolves the initial "V" shaped network to the minimum transportation network.
Remark 3.7. The language of "may choose" in Lemma 3.5 stresses the non-uniqueness of the flow due to perhaps having choices in the creation of the step minimizing sequence. This particularly happens with high levels of symmetry. For example, there are multiple global optimal networks connecting the source and sinks in To find the bifurcation point (x(t), 0) then of T t for t > 0, choose so that h ≤ t < ( + 1)h for each h > 0. Rearranging Equation 3.5 and summing, we have j=1 then, let h → 0: Hence, the rate of zipping is given by x(t) which solves the following first order, non-linear ordinary differential equation:

Two sources; two sinks
Suppose that µ + = mδ (−w/2, /2) + mδ (w/2, /2) and µ − = mδ (−w/2,− /2) + mδ (w/2,− /2) , for m, w, ∈ R + . That is, the sources and sinks lie on the vertices of a rectangle of width w and length . There are two possible optimal transportation networks for α ∈ [0, 1] depending on the configuration of µ + and µ − : a branching path (denoted P b ) or one composed of two straight edges (denoted P s ). For α ∈ (0, 1], we will now consider the flow acting on two different initial paths from µ + to µ − : one with an "X" shape and the other with a "| |" shape. Both are drawn in Figure 3.3. The initial network having "| |" shape demonstrates that our minimizing flow merely finds local optimal solutions similar to other minimizing or gradient descent flows. However, we will see that the "X" shaped initial network always flows to the global optimal solution regardless of the configuration of the boundary. Example 3.9. Suppose T 0 is a transport path directly connecting each source to a sink in a "| |" shaped path. This network fails to evolve under the flow regardless of the value of . That is to say, for every h > 0, each T h j must be two parallel, directed line segments; and thus, T t for almost every t > 0 will stay two parallel directed line segments. (See Figure 3.4 for a representation of the two dimensional flow S whose slices are these T t .) Hence, the flow finds a local minimum in this case.
One may see this by examining the functional in Equation (2.1) in Section 2. With any perturbation of the previous path T h j−1 , one would increase length of the directed segments and thus, the M α mass of the path, while simultaneously forcing a positive Fill α term. Hence, for every fixed h > 0, the discrete sequence must only be comprised of T 0 . Example 3.10. Although the above initial network merely flowed to local optimal network, an initial path with an "X" shape flows to the global optimal solution regardless of the boundary configuration. We will show how the flow on this initial network always includes the necessary topology change to flow to the global minimum in two specific boundary configurations.
Suppose we instead begin with an "X" shaped transport path T 0 from µ + to µ − . If = w, then we may choose the minimizer for every h > 0 and j > 0 having a "> <" shape as in Figure  3.5b to be T h j , and hence, construct the flow so that every T t has this same shape for almost every t > 0. The path stabilizes under the flow once it reaches P s , which is the optimal network. In this case, the flow changes the topology of the network as needed to find the global minimum since we lose an interior vertex in the breaking process. (See Figure 3.5b for a representation of the two dimensional flow S whose slices are these T t .) If = 4w, then we may choose the minimizer for every h > 0 and j > 0 having a branching structure as in Figure 3.5a to be T h j , and hence, construct the flow so that every T t has this same branching structure for almost every t > 0. In this case, the path simultaneously zips up from both ends until reaching the angles of Here is a representation of the constructed two dimensional flow S for Example 3.9. We cut it off at some finite time.
the optimal network P b . Since we gain an interior vertex through the zipping process, the flow again changes the topology of the initial network as needed to find the global minimum. (See Figure 3.5a for a representation of the two dimensional flow S whose slices are these T t .) Proof. Using similar convexity arguments as those used in Example 3.5, we may show that the paths must zip up along either axis due to the symmetry of µ + and µ − . Zipping along the y-axis would result in an edge with associated weight 2m, while zipping along the x-axis may be thought of as resulting in an edge with associated weight 0. See Figure 3.5.  Let's see how T x reduces cost in the = w case and T y reduces cost in the = 4w case: If = w, then G 1 (T x ) < G 1 (T y ) for any > 0. Hence, the initial path breaks apart and flows to P s , as desired. See Figure 3.6.
On the other hand, if = 4w, then G 1 (T x ) > G 1 (T y ) for sufficiently small > 0. Thus, the initial path must zip up along the y-axis and flow to P b . See  (The rate of zipping will affect the shape.) We cut it off at the time when it achieves the optimal shape P s . Remark 3.11. While we only considered cases when zipping occurs, one may show similar results for paths that must unzip instead. Our constructed flow's ability to alter the topology of the transport network in cost saving ways is due to our choice of functional G defined in Equation 2.1 determining the chains in the step minimizing sequences.

Open Questions
Constructing optimal paths between atomic measures via current techniques proves computationally difficult often due to the vast number of possible topologies. Since this M α mass reducing flow is able to change the topology of the initial network, one hopes that the flow will overcome this difficulty, solving the Gilbert transport optimization problem. Unfortunately, Example 3.9 shows that the flow generally finds merely local minima. However, Example 3.10 leads us to conjecture that we may hope to pick a portfolio of initial paths with at least one flowing to an optimal transportation network. We hope that the topology will evolve as needed to move towards the global optimal transport network by "breaking apart" and zipping up when starting with similar "coned" initial networks as that in Example 3.10.
Conjecture 4.1. For atomic measures µ + and µ − , an initial starting transport path composed of "coned" networks (having "hubs" for every anticipated connected component of an optimal path) will flow to an optimal transport network from µ + to µ − . Here is a representation of the constructed, two dimensional flow S when = 4w. (The rate of zipping will affect the shape.) We cut it off at the time when it achieves the optimal shape P b .
Because of the results shown in the above examples, we additionally hope that existing discrete computational methods may be enhanced by incorporating the ideas in functional G in such ways that allow for detection of topological modifications of an approximation for cost reduction.