On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

Abstract

A Gérard pour ses 2 6 et 2 6 +1-èmes anniversaires.Abstract This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gérard Cohen influenced important parts of my research.The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalizing the notion of separating systems in the context of metric convexity.The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry -sets of points forming only acute angles -who seem not to be aware of each other, while they are working on precisely the same problem! Of course, these three themes are closely intertwined.