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 characterize 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.

反范数是凸锥上的凹非负齐次泛函。结果表明，如果圆锥是多面体，则每个反范数都从圆锥内部有唯一的连续延伸。凸分析中对偶理论的主要事实，特别是 Fenchel-Moreau 定理，被推广到反范数。然而，它表明反规范的对偶关系是不连续的。在每个维度中，在正轴上有无限多个自对偶反规范，特别是有无限多个自极多面体。对于二维情况，我们将它们全部表征。更高维度的分类作为一个悬而未决的问题。应用于线性动力系统、随机矩阵乘积的 Lyapunov 指数、非负矩阵的较低谱半径，