Upper semicontinuity of the lamination hull^★

¹ School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
² Department of Mathematics, University of Illinois, Urbana IL 61801, USA

^* Corresponding author: terence2@illinois.edu

Received: 12 October 2016
Received in final form: 18 March 2017
Accepted: 21 April 2017

Abstract

Let  K ⊆ ℝ^2×2 be a compact set, let K^rc be its rank-one convex hull, and let L (K) be its lamination convex hull. It is shown that the mapping K ↦ L̅(K̅) is not upper semicontinuous on the diagonal matrices in ℝ^2×2, which was a problem left by Kolář. This is followed by an example of a 5-point set of 2 × 2 symmetric matrices with non-compact lamination hull. Finally, another 5-point set K is constructed, which has L (K) connected, compact and strictly smaller than K^rc.