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 α > 1 the function Nᵅ is strictly convex (analytic characterization).

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Keywords

  • Minkowski norms
  • affine geometry
  • quasi-convexity
  • gauge functions
  • topological vector spaces
  • strict sub-convexity

Mathematics Subject Classification

52A07; 52A05

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Suggested citation

S. Simon, P. Verovic. “When is a Minkowski Norm Strictly Sub-Convex?.” Journal of Convex Analysis 31 (2024), No. 1, 139–178. https://doi.org/10.68381/jca31010

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.