A Geometric Proof of Ryll-nardzewski's Fixed Point Theorem by I. Namioka and E. Asplund

In [4], Ryll-Nardzewski gave what he called an 'old-fashioned' proof of his famous fixed point theorem. The purpose of the present note is to give an even more old-fashioned proof of the fixed point theorem. In fact, our proof uses nothing more than a category argument and the classical Krein-Milman theorem. Our terminology and notation shall be those of Kelley, Namioka et al. [2]. The following geometric lemma is essential to our proof of Ryll-Nardzewski's fixed point theorem. In case the space E and the pseudo-norm p in the lemma are a Banach space and its norm respectively, the lemma is an easy consequence of Lindenstrauss' work [3].