Antinorms on cones: duality and applications

TL;DR

This paper introduces antinorms on convex cones, explores their duality properties, and applies these concepts to various areas including linear dynamical systems and convex trigonometry, revealing new theoretical insights and classifications.

Contribution

It generalizes duality theory to antinorms on polyhedral cones, characterizes self-dual antinorms in two dimensions, and applies these findings to multiple mathematical fields.

Findings

Duality relation for antinorms is discontinuous.

Infinitely many self-dual antinorms exist on the positive orthant.

Complete characterization of self-dual antinorms in two dimensions.

Abstract

An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality theory in convex analysis, in particular, the Fenchel-Moreau theorem, are generalized to antinorms. However, it is shown that the duality relation for antinorms is discontinuous. In every dimension there are infinitely many self-dual antinorms on the positive orthant and, in particular, infinitely many autopolar polyhedra. For the two-dimensional case, we characterise them all. The classification in higher dimensions is left as an open problem. Applications to linear dynamical systems, to the Lyapunov exponent of random matrix products, to the lower spectral radius of nonnegative matrices, and to convex trigonometry are considered.