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

\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 set-valued map $p\mapsto M_p$ is constant on the intervals $[1,2),\dots,[q-1,q),[q,\infty)$ 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.