A Krasnosel'skii-type theorem for paths of bounded length (Q1816483)

Let \(S\) be a nonempty closed, simply connected set in the plane, and let \(\alpha>0\). If every three points of \(S\) see a common point of \(S\) via paths of length at most \(\alpha\), then for some points \(s_0\) of \(S\), \(s_0\) sees each point of \(S\) via such a path. That is, \(S\) is starshaped via paths of length at most \(\alpha\).