  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.
