Steinhaus-type property for a boundary of a convex body

TL;DR

This paper characterizes the Steinhaus-type property for neighborhoods on the boundary of convex bodies, linking it to boundary flatness and implications for the continuity of additive and mid-convex functions.

Contribution

It establishes a precise condition under which neighborhoods on convex body boundaries have the Steinhaus-type property, connecting geometric boundary features with functional continuity.

Findings

Neighborhoods with the Steinhaus-type property correspond to non-flat boundary points.

Additive and mid-convex functions bounded above on such neighborhoods are continuous.

The Steinhaus-type property is characterized by the non-flatness of boundary points.

Abstract

We show that if $U\subset\partial A$ is a neighbourhood of a point $x_0\in\partial A$ of the boundary of a convex body $A$ then it has the so-called Stainhaus-type property (the interior of $(U+U)$ is nonempty) if and only if $x_0$ is not a point of flatness of the boundary~$\partial A$. This implies that additive functions as well as mid-convex functions, bounded above on~$U$, are continuous.