
Journal of Convex Analysis 14 (2007), No. 2, 319344 Copyright Heldermann Verlag 2007 Zeros of the Polyconvex Hull of Powers of the Distance and sPolyconvexity Miroslav Silhavy Dip. di Matematica, Universitą di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy Permanent Address: Mathematical Institute, Academy of Sciences, Zitnį 25, 115 67 Prague 1, Czech Republic silhavy@math.cas.cz [Abstractpdf] \def\dist{\mathop{\rm dist}\nolimits} \def\P{{\mathsf P}} Let $\dist_K$ 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 $\dist_K^p$ where $1\leq p<\infty$. It is shown that the setvalued map $p\mapsto M_p$ is constant on the intervals $[1,2),\dots,[q1,q),[q,\infty)$ 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 $\P^sK$ of $K$ to be defined below, where $\P^1K$ is the convex hull and $\P^qK$ the standard polyconvex hull. As an illustration, $\P^sSO(n)$ are evaluated for all $s$ if $1\leq n\leq 4$, and for $n$ arbitrary if $n\geq s>n/2$ and/or $s=1$. In the remaining cases only bounds are obtained. Keywords: Semiconvexity, polyconvexity, polyconvex hulls, rotational invariance. MSC: 49J45; 74B20 [ Fulltextpdf (235 KB)] for subscribers only. 