On the Strong Quasiconvexity of Norms and Distance Functions

Abstract

This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this property for general norms remains open. We establish necessary and sufficient conditions for a norm function to be strongly quasiconvex on a convex set. We also initiate the study of the strong quasiconvexity of distance functions. Our results provide new insights into the geometric properties of norm and distance functions and extend several existing results in the literature.