Steinhaus-type property for a boundary of a convex body

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.