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Near-optimal generalisations of a theorem of Macbeath. (English) Zbl 1359.52008

Mayr, Ernst W. (ed.) et al., 31st international symposium on theoretical aspects of computer science, STACS’ 14, Lyon, France, March 5–8, 2014. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik (ISBN 978-3-939897-65-1). LIPIcs – Leibniz International Proceedings in Informatics 25, 578-589 (2014).
Summary: The existence of Macbeath regions is a classical theorem in convex geometry [A. M. Macbeath, Ann. Math. (2) 56, 269–293 (1952; Zbl 0047.04903)]. We refer the reader to the survey of I. Bárány [Lect. Notes Math. 1892, 77–118 (2007; Zbl 1123.60006)] for several applications. 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 espilon-nets for geometric set systems.
For the entire collection see [Zbl 1294.68025].

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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