A Helly-type theorem for semi-monotone sets and monotone maps

Saugata Basu, Andrei Gabrielov, Nicolai Vorobjov

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Abstract

We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or globally subanalytic sets. A monotone map is a multi-dimensional generalization of a usual univariate monotone continuous function on an open interval, while the closure of the graph of a monotone map is a generalization of a compact convex set. In a particular case of an identically constant function, such a graph is called a semi-monotone set. Graphs of monotone maps are, generally, non-convex, and their intersections, unlike intersections of convex sets, can be topologically complicated. In particular, such an intersection is not necessarily the graph of a monotone map. Nevertheless, we prove a Helly-type theorem, which says that for a finite family of subsets of {Mathematical expression}, if all intersections of subfamilies, with cardinalities at most {Mathematical expression}, are non-empty and graphs of monotone maps, then the intersection of the whole family is non-empty and the graph of a monotone map.
Original languageEnglish
Pages (from-to)857-864
Number of pages8
JournalDiscrete & Computational Geometry
Volume50
Issue number4
Early online date17 Sep 2013
DOIs
Publication statusPublished - Dec 2013

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Keywords

  • Monotone maps, Helly's theorem

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