Topological properties of closed weakly $m$-convex sets

TL;DR

This paper explores the topological and convexity properties of weakly m-convex sets in Euclidean space, establishing conditions for their connectedness and constructing examples for various dimensions and convexity levels.

Contribution

It provides new insights into the structure of weakly m-convex sets, including their connected components and nonconvexity points, extending the theory of convexity in higher dimensions.

Findings

Any closed, weakly (n-1)-convex set with non-empty nonconvexity points has at least three components.

The interior of a closed, weakly 1-convex set with finitely many components in the plane is weakly 1-convex.

Constructed examples of weakly m-convex domains and sets with non-empty nonconvexity points for all relevant dimensions.

Abstract

The present work considers the properties of generally convex sets in the $n$-dimensional real Euclidean space $\mathbb{R}^n$, $n>1$, known as weakly $m$-convex, $m=1,2,\ldots,n-1$. An open set of $\mathbb{R}^n$ is called weakly $m$-convex if for any boundary point of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of $\mathbb{R}^n$ is called weakly $m$-convex if it is approximated from the outside by a family of open weakly $m$-convex sets. A point of the complement of a set of $\mathbb{R}^n$ to the whole space is called an $m$-nonconvexity point of the set if any $m$-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly $(n-1)$-convex set in $\mathbb{R}^n$ with non-empty set of $(n-1)$-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly $1$-convex set with a finite number of components in the plane is weakly $1$-convex. Weakly $m$-convex domains and closed connected sets in $\mathbb{R}^n$ with non-empty set of $m$-nonconvexity points are constructed for any $n\ge 3$ and any $m=1,2,\ldots,n-2$.