On the minimal distance between two surfaces

TL;DR

This paper revisits and refines previous results on the minimal distance between two surfaces, emphasizing the usefulness of the Gao-Strang total complementary function for establishing necessary conditions.

Contribution

It modifies earlier findings by incorporating Voisei and Zalinescu's perspectives and highlights the importance of the Gao-Strang function in the problem.

Findings

Refined conditions for minimal distance between surfaces

Demonstrated the utility of the Gao-Strang total complementary function

Clarified the relationship between previous and new results

Abstract

This article revisits previous results presented in Optimization which were challenged later by Voisei and Zalinescu (V-Z) in the same journal. We aim to use the points of view of V-Z to modify the original results and highlight that the consideration of the so called Gao-Strang total complementary function is indeed quite useful for establishing necessary conditions for solving this problem.