Rectifiable Curves in Proximally Smooth Sets

Abstract

We study shortest curves in proximally smooth subsets of a Hilbert space. We consider a 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 space -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.