Shortest curves in proximally smooth sets: existence and uniqueness

TL;DR

This paper establishes the existence and uniqueness of shortest curves within proximally smooth sets in Hilbert spaces, providing a geometric construction and demonstrating the tightness of the length bound.

Contribution

It introduces a simple geometric algorithm for constructing shortest curves in proximally smooth sets and proves their uniqueness, extending classical results to infinite-dimensional spaces.

Findings

Existence of a shortest curve connecting two points in proximally smooth sets.

A geometric algorithm for constructing such shortest curves.

The length bound is tight and attained on Euclidean spheres.

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 $R.$