Shortest Curves in Proximally Smooth Sets: Existence and Uniqueness

We study shortest curves in proximally smooth subset of a finite or infinite dimensional Hilbert space. We consider an R-proximally smooth set A in a Hilbert space with points a and b satisfying |a - b| < 2R. We provide a simple geometric algorithm of constructing a curve inside A connecting a and b whose length is at most 2R arcsin |a - b|/2R, which corresponds to the shortest curve inside the model set - a Euclidean sphere of radius R passing through a and b. Using this construction, we show that there exists a unique shortest curve inside A connecting a and b. This result is tight since two points of A at distance 2R are not necessarily connected in A; the bound on the length cannot be improved since the equality is attained on the Euclidean sphere of radius R.