On the local structure of rank-one convex hulls

Abstract

In this note we prove that ifKKis a compact set ofm×nm\times nmatrices containing an isolated pointXXwith no rank-one connection into the convex hull ofK∖{X}K\setminus \{X\}, then the rank-one convex hull separates as\[Krc=(K∖{X})rc∪{X}.K^{rc}=\bigl (K\setminus \{X\}\bigr )^{rc}\cup \{X\}.\]This is an extension of a result of P. Pedregal, which holds for2×22\times 2matrices.