Finitely starlike sets whose F-stars have positive measure

Abstract

Let\(S \subseteq R^2\) and assume that there is a countable collection of lines {L i : 1 ≤ i} such that (int cl S)\( \sim S \subseteq \cup \{ L_i :1 \leqslant i\}\) and ((int cl S) ∼S) ∩Li has one-dimensional Lebesgue measure zero, 1 ≤i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with ‘two-dimensional’ replaced by ‘one-dimensional’. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.