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
Stephane.Simon@univ-smb.fr
Patrick Verovic
Université Savoie Mont Blanc, CNRS, LAMA, Chambéry, France
Patrick.Verovic@univ-smb.fr
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.