A characterization of the duality mapping for convex bodies
For \(d \geq 2\), let \(\mathbb{R}^d\) be \(d\)-dimensional euclidean space, equipped with the standard scalar product \(\langle \cdot,\cdot \rangle\), and let \(\mathcal{K}_{(0)}^d\) denote the set of convex bodies in \(\mathbb{R}^d\) containing the origin \(0\) in their interiors. As usual, the dual or polar of \(K \in \mathcal{K}_{(0)}^d\) is defined by \[ K^* := \{x \in \mathbb{R}^d : \langle x,y \rangle \leq 1 \text{ for all } y \in K\}. \] Writing \(A \vee B := \text{conv}(A \cup B)\), the authors characterize polarity as follows: if \(\psi: \mathcal{K}_{(0)}^d \to \mathcal{K}_{(0)}^d\) is a mapping satisfying (1) \(\psi(K \cap L) = \psi(K) \vee \psi(L)\) and (2) \(\psi(K \vee L) = \psi(K) \cap \psi(L)\) for all \(K,L\), then either \(\psi\) is constant or there is a linear transformation \(g \in GL(d)\) such that \(\psi(K) = gK^*\) for all \(K\). This generalizes the special case of centrally symmetric bodies proved by \textit{P. M. Gruber} [Abh. Math. Semin. Univ. Hamb. 62, 179--189 (1992; Zbl 0779.52006)]. The authors point out that no continuity assumption is needed. As a corollary, if such a mapping \(\psi\) is involutory, then \(\psi(K) = gK^*\) for some self-adjoint \(g \in GL(d)\).
scientific article; zbMATH DE number 5565917
scientific article