When is a Minkowski Norm Strictly Sub-Convex?

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).