Admissible kernels for starshaped sets

Abstract

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