When is a Minkowski norm strictly sub-convex?

Stéphane Simon; Patrick Verovic

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arXiv · 2206.01016

When is a Minkowski norm strictly sub-convex?

Stéphane Simon·Patrick Verovic

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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).

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Stéphane Simon, Patrick Verovic. 2022-06-02. When is a Minkowski norm strictly sub-convex?. https://arxiv.org/abs/2206.01016

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