Talk

Zeroes of the polyconvex hull of powers of the distance and s-polyconvexity

  • Miroslav Silhavy (Czech Academy of Science)
A3 01 (Sophus-Lie room)

Abstract

Let \distK be the distance from a compact set K\matrices in the space of m×n matrices. This note determines the set Mp\matrices of zeroes of the polyconvex hull of \distKp where 1p<. It is shown that the set-valued function pMp is constant on the intervals [1,2),,[q1,q),[q,) where q:=min\dfsetm,n, while at p=1,,q the set Mp generally jumps down discontinuously. The values Ms,s=1,,q, at the beginnings of intervals of constancy are characterized as s-polyconvex hulls \PsK of K to be defined below, where \P1K is the convex hull and \PqK the standard polyconvex hull. As an illustration, \PsSO(n) are evaluated for all s if 1n4, and for n arbitrary if ns>n/2 and/or s=1. In the remaining cases only bounds are obtained. A surprising consequence is that the quasiconvex hull of \distSO(n)p is not polyconvex if 1p<n.</p>

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