A Brouwer type theorem (Q1262571)

The well known and very plausible conjecture, that any contractible compact set of \({\mathbb{R}}^ n\) has the fixed point property was disproved by \textit{S. Kinoshita} in Fundam. Math. 40, 96-98 (1953; Zbl 0053.125). He gave a two dimensional contractible continuum without the fixed point property. The aim of the present note is to prove the fixed point teorem to the ``star-like'' compact sets of \({\mathbb{R}}^ 2\).