Shortest curves in proximally smooth sets: existence and uniqueness

Abstract

We study shortest curves in proximally smooth subsets of a Hilbert space. We consider an $R$-proximally smooth set $A$ in a Hilbert space with points $a$ and $b$ satisfying $\left|{a-b}\right| < 2R.$ We provide a simple geometric algorithm of constructing a curve inside $A$ connecting $a$ and $b$ whose length is at most $2R \arcsin\frac{\left|{a-b}\right|}{2R},$ 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 $R.$