When is a Minkowski norm strictly sub-convex?

TL;DR

This paper characterizes when a Minkowski norm has strictly convex sublevel sets, providing geometric and analytic conditions involving continuity and the convexity of powers of the norm.

Contribution

It offers two complete characterizations of Minkowski norms with strictly convex sublevel sets, linking geometric and analytic properties.

Findings

Equivalence of strict convexity of sublevel sets to continuity and boundary chord conditions.

Equivalence of strict convexity to the convexity of N^α for α > 1.

Provides simple, complete criteria for strict sub-convexity of Minkowski norms.

Abstract

The aim of this paper is to give two complete and simple characterizations of Minkowski norms N on an arbitrary topological real vector space such that the sublevel sets of N are strictly convex. We first show that this property is equivalent to the continuity of N together with the fact that any open chord between two points of the boundary of the sublevel set N^{-1}([0, 1)) lies inside that set (geometric characterization). On the other hand, we prove that this is also the same as saying that N is continuous and that for an arbitrary real number $\alpha$ > 1 the function N^$\alpha$ is strictly convex (analytic characterization).