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

Let ${\text{dist}}_K$ be the distance from a compact set $K\subset \text{matrices}$ in the space of $m\times n$ matrices. This note determines the set $M_p\subset \text{matrices}$ of zeroes of the polyconvex hull of ${\text{dist}}_K^p$ where $1\le p<\mathrm{\infty }.$ It is shown that the set-valued function $p\mapsto M_p$ is constant on the intervals $[1,2),\dots ,[q-1,q),[q,\mathrm{\infty })$ where $q:=min\text{dfset}m,n,$ 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 ${\text{P}}^{s}K$ of $K$ to be defined below, where ${\text{P}}^{1}K$ is the convex hull and ${\text{P}}^{q}K$ the standard polyconvex hull. As an illustration, ${\text{P}}^{s}SO(n)$ 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. A surprising consequence is that the quasiconvex hull of ${\text{dist}}_{SO(n)}^p$ is not polyconvex if $1\le p<n.$</p>