SCHWEITZER 1974 Problem 9

Let $A$ be a closed and bounded set in the plane, and let $C$ denote the set of points at a unit distance from $A$ . Let $p \in C$ , and assume that the intersection of $A$ with the unit circle $K$ centered at $p$ can be covered by an arc shorter that a semicircle of $K$ . Prove that the intersection of $C$ with a suitable neighborhood of $p$ is a simple arc which $p$ is not an endpoint.

M. Bognar