Semi-monotone sets

TL;DR

This paper introduces semi-monotone sets, a generalization of convex sets within o-minimal structures, and proves that all such sets are topologically regular cells, revealing their geometric and combinatorial properties.

Contribution

The paper establishes that every semi-monotone set is a topological regular cell, expanding understanding of their structure and properties.

Findings

Semi-monotone sets are topological regular cells.

They generalize convexity in o-minimal structures.

Semi-monotone sets have notable geometric and combinatorial properties.

Abstract

A coordinate cone in R^n is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is a defnable in an o-minimal structure over the reals, open bounded subset of R^n such that its intersection with any translation of any coordinate cone is connected. This can be viewed as a generalization of the convexity property. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone set is a topological regular cell.