Admissible kernels for starshaped sets

Abstract:

Steven Lay has posed the following interesting question: If $D$ is a convex subset of ${{\mathbf {R}}^d}$, then is there a starshaped set $S \ne D$ in ${{\mathbf {R}}^d}$ whose kernel is $D$? Thus the problem is that of characterizing those convex sets which are admissible as the kernel of some nonconvex starshaped set in ${{\mathbf {R}}^d}$. Here Lay’s problem is investigated for closed sets, and the following results are obtained: If $D$ is a nonempty closed convex subset of ${{\mathbf {R}}^2}$, then $D$ is the kernel of some planar set $S \ne D$ if and only if $D$ contains no line. If $D$ is a compact convex set in ${{\mathbf {R}}^d}$, then there is a compact set $S \ne D$ in ${{\mathbf {R}}^d}$ whose kernel is $D$.