A Krasnosel'skii theorem for open finitely starlike sets in R 3

Let S be an open set in R ³ such that: (1) whenever three points x, y, z in S see each other via S, then conv{x, y, z} S, and (2) every seven points in S see via S a common point. Then S is finitely starlike. The proof uses the topological version of Helly's theorem.