2012/02/06 by Basu, Saugata, Gabrielov, Andrei, Vorobjov, Nicolai
#03C64 #52A30 #52A35 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1202.1198
We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or subanalytic sets. A \em monotone map is a multi-dimensional generalization of a usual univariate monotone function, 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 \em 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 \Realn, 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.