Journal Home Page

Cumulative Index

List of all Volumes

Complete Contents
of this Volume

Previous Article

Next Article

Journal of Convex Analysis 31 (2024), No. 1, 139--178
Copyright Heldermann Verlag 2024

When is a Minkowski Norm Strictly Sub-Convex?

Stéphane Simon
Université Savoie Mont Blanc, CNRS, LAMA, Chambéry, France

Patrick Verovic
Université Savoie Mont Blanc, CNRS, LAMA, Chambéry, France

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 $Nα is strictly convex (analytic characterization).

Keywords: Minkowski norms, affine geometry, quasi-convexity, gauge functions, topological vector spaces, strict sub-convexity.

MSC: 52A07; 52A05.

[ Fulltext-pdf  (243  KB)] for subscribers only.