Zeros of the Polyconvex Hull of Powers of the Distance and s-Polyconvexity

M. Silhavy

Journal of convex analysis, Tome 14 (2007) no. 2, pp. 319-344.

Résumé

\def\dist{\mathop{\rm dist}\nolimits} \def\P{{\mathsf P}} Let ${\text{dist}}_K$ be the distance from a compact set $K\subset {\mathbb{M}}^{m\times n}$ in the space of $m\times n$ matrices. This note determines the set $M_p\subset {\mathbb{M}}^{m\times n}$ of zeros of the polyconvex hull of ${\text{dist}}_K^p$ where $1\le p\mathrm{\infty }$. It is shown that the set-valued map $p\mapsto M_p$ is constant on the intervals $[1,2),\dots ,[q-1,q),[q,\mathrm{\infty })$ where $q:=min\{m,n\}$, while at $p=1,\dots ,q$ the set $M_p$ generally jumps down discontinuously. The values $M_s$, $s=1,\dots ,q$, at the beginnings of intervals of constancy are characterized as $s$-polyconvex hulls ${\mathbb{P}}^{s}K$ of $K$ to be defined below, where ${\mathbb{P}}^{1}K$ is the convex hull and ${\mathbb{P}}^{q}K$ the standard polyconvex hull. As an illustration, ${\mathbb{P}}^{s}SO(n)$ are evaluated for all $s$ if $1\le n\le 4$, and for $n$ arbitrary if $n\ge s>n/2$ and/or $s=1$. In the remaining cases only bounds are obtained.

Détail

Classification : 49J45, 74B20
Mots-clés : Semiconvexity, polyconvexity, polyconvex hulls, rotational invariance