On semiconvexity and weak semiconvexity

TL;DR

This paper investigates properties of m-semiconvex and weakly m-semiconvex sets in Euclidean space, revealing that certain weakly 1-semiconvex open sets with smooth boundaries in the plane have at least four components.

Contribution

It provides new insights into the structure of weakly m-semiconvex sets, especially demonstrating the minimal number of components for specific weakly 1-semiconvex sets.

Findings

Weakly 1-semiconvex open sets with smooth boundary in the plane have at least four components.

The paper characterizes the properties of m-semiconvex and weakly m-semiconvex sets in Euclidean space.

Establishes conditions under which these sets are connected or disconnected.

Abstract

Properties of two classes of generally convex sets in the n-dimentional real Euclidean space, called m-semiconvex and weakly m-semiconvex, 1<=m<n, are investigated in the present work. In particular, it is established that an open set with smooth boundary in the plan which is weakly 1-semiconvex but not 1-semiconvex consists minimum of four simply connected components.