A Helly-Type Theorem for Semi-monotone Sets and Monotone Maps |
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Authors: | Saugata Basu Andrei Gabrielov Nicolai Vorobjov |
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Institution: | 1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA 2. Department of Computer Science, University of Bath, Bath, BA2 7AY, England, UK
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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 $\mathbb{R }^n$ , if all intersections of subfamilies, with cardinalities at most $n+1$ , 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. |
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