A representation formula for weakly compact starshaped sets

Abstract

Let S be a nonconvex weakly compact and weakly connected subset of a real locally convex topological linear space L and D a relatively weakly open subset of S containing the set Inc$_{w}S$ of local nonconvexity points of S with respect to the weak topology. It is proved that ker$S=\bigcap{\textrm{clconv} S_{z}: z ∈ D ∩ \,\textrm{reg} S}$, where regS denotes the set of regular points of S and $S_{z} = {s ∈ S: z \textrm{ is visible from} s \textrm{ via} S}$. This substantially stregthens a recent result of Stavrakas in which the intersection above was taken over the whole set regS. The intersection formula is shown to hold also for a nonconvex connected weakly compact subset S of L with D being a relatively weakly open subset of S containing the set IncS of local nonconvexity points of S.