Autopolar conic bodies and polyhedra

TL;DR

This paper explores the properties of antinorms, which are concave analogues of norms defined on cones, focusing on their level sets called conic bodies and polyhedra, and classifies self-dual cases especially in two dimensions.

Contribution

It introduces the concept of autopolar conic bodies and polyhedra, proving the existence of infinitely many such families in the cone R^d_+ and classifying self-dual antinorms in two dimensions.

Findings

Existence of infinitely many autopolar conic bodies and polyhedra in R^d_+

Complete classification of self-dual antinorms for d=2

Counterexamples to self-duality in dimensions d≥3

Abstract

An antinorm is a concave analogue of a norm. In contrast to norms, antinorms are not defined on the entire space $R^d$ but on a cone $K\subset R^d$. They are applied in the matrix analysis, optimal control, and dynamical systems. Their level sets are called conic bodies and (in case of piecewise-linear antinorms) conic polyhedra. The basic facts and notions of the "concave analysis" of antinorms such as separation theorems, duality, polars, Minkowski functionals, etc., are similar to those from the standard convex analysis. There are, however, some significant differences. One of them is the existence of many self-dual objects. We prove that there are infinitely many families of autopolar conic bodies and polyhedra in the cone $K=R^d_+$. For $d=2$, this gives a complete classification of self-dual antinorms, while for $d\ge 3$, there are counterexamples.