Abstract:
An antinorm in a linear space is a concave analogue of a norm. In contrast to norms, antinorms are not defined on the whole space $\mathbb{R}^d$ but on a cone $K\subset \mathbb{R}^d$. They are applied to functional analysis, optimal control and dynamical systems. Level sets of antinorms are called conic bodies and (in the 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, and so on, are similar to the ones in 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 $K=\mathbb{R}^d_+$. For $d=2$ this gives a complete classification of self-dual antinorms, while for $d\ge 3$ there are counterexamples.
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