Near-Optimal Generalisations of a Theorem of Macbeath

International audienceThe existence of Macbeath regions is a classical theorem in convex geometry ("A Theorem on non-homogeneous lattices'', Annals of Math, 1952). We refer the reader to the survey of I. Barany for several applications~\cite{B07}. Recently there have been some striking applications of Macbeath regions in discrete and computational geometry. In this paper, we study Macbeath's problem in a more general setting, and not only for the Lebesgue measure as is the case in the classical theorem. We prove near-optimal generalizations for several basic geometric set systems. The problems and techniques used are closely linked to the study of epsilon-nets for geometric set systems