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

TL;DR

This paper proves a Helly-type theorem for intersections of semi-monotone sets and graphs of monotone maps within o-minimal structures, extending convex intersection properties to more complex, non-convex sets.

Contribution

It establishes a Helly-type result for semi-monotone sets and monotone maps, generalizing classical convex intersection theorems to non-convex, topologically complex sets.

Findings

Intersections of certain families of monotone map graphs are non-empty and monotone.

Finite intersection conditions guarantee the entire intersection retains the monotone map structure.

The theorem extends Helly's theorem to semi-monotone and monotone map sets in o-minimal structures.

Abstract

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 $\Real^n$, 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.