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

Let $\dist_K$ be the distance from a compact set $K\subset\matrices$ in the space of $m\times n$ matrices. This note determines the set $M_p\subset\matrices$ of zeroes of the polyconvex hull of $\dist_K^p$ where $1\leq p<\infty.$ It is shown that the set-valued function $p\mapsto M_p$ is constant on the intervals $[1,2),\dots,[q-1,q),[q,\infty)$ where $q:=\min\dfset{m,n},$ 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 $1\leq n\leq4,$ and for $n$ arbitrary if $n\geq s>n/2$ and/or $s=1.$ In the remaining cases only bounds are obtained. A surprising consequence is that the quasiconvex hull of $\dist_{SO(n)}^p$ is not polyconvex if $1\leq p<n.$</p>