Monovex Sets

TL;DR

This paper investigates monovex sets in Euclidean spaces, proving that open and closed monovex sets are contractible and providing a counterexample for nonopen, nonclosed sets, thus revealing key topological properties.

Contribution

The paper establishes contractibility of open and closed monovex sets and presents a counterexample for nonopen, nonclosed monovex sets, advancing understanding of their topological structure.

Findings

Open and closed monovex sets are contractible.

Counterexample of a nonopen, nonclosed monovex set that is not contractible.

Additional properties of monovex sets are revealed.

Abstract

A set $A$ in a finite dimensional Euclidean space is \emph{monovex} if for every two points $x,y \in A$ there is a continuous path within the set that connects $x$ and $y$ and is monotone (nonincreasing or nondecreasing) in each coordinate. We prove that every open monovex set as well as every closed monovex set is contractible, and provide an example of a nonopen and nonclosed monovex set that is not contractible. Our proofs reveal additional properties of monovex sets.