Finitely starlike sets whose F-stars have positive measure (Q582579)

Krasnoselskij's classical theorem characterizes compact starshaped sets in Euclidean spaces. Without the assumption of compactness the corresponding results become much more intricate. The present article describes, among other things, such a situation in the Euclidean plane, with much zeal and ingenuity. Given a planar set S, we say that X sees Y via S, if every line connecting some point of X to some point of Y belongs to S. Let T be the interior of the closure of S. If \(T\setminus S\) is small, in a sense to be made precise, then every finite subset X of S sees some subset L(X) with positive area, via S, provided that every fourpointed subset of S does.