Steinhaus-type property for the boundary of a convex body (Q2314860)

The main result of this note asserts that a neighborhood $U$ of a point $x_0$ that lies on the boundary of a convex body $A$ has the Steinhaus-type property $int (U +U) \neq\emptyset$ if and only if $x_0$ is not a point of flatness of the mentioned boundary. A consequence of this result is that additive functions bounded above on $U$ are continuous. Some examples and open problems close the paper.