TY - JOUR

T1 - A Helly-type theorem for semi-monotone sets and monotone maps

AU - Basu, Saugata

AU - Gabrielov, Andrei

AU - Vorobjov, Nicolai

PY - 2013/12

Y1 - 2013/12

N2 - 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.

AB - 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.

KW - Monotone maps, Helly's theorem

UR - http://www.scopus.com/inward/record.url?scp=84883696067&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s00454-013-9540-y

U2 - 10.1007/s00454-013-9540-y

DO - 10.1007/s00454-013-9540-y

M3 - Article

SN - 0179-5376

VL - 50

SP - 857

EP - 864

JO - Discrete & Computational Geometry

JF - Discrete & Computational Geometry

IS - 4

ER -