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Minkowski endomorphisms. (English) Zbl 1372.52005

A Minkowski endomorphism is an additive continuous \(SO(n)\)-equivariant and translation invariant map from \(\Psi\) from \(\mathcal{K}^n\) to \(\mathcal{K}^n\), the set of nonempty compact convex sets in \(\mathbb R^n\). First, the author recalls a known result, which states that if \(\Psi\) is a Minkowski endomorphism, then \(h_{\Psi K}=h_K*\nu\) for all \(K\in\mathcal{K}^n\), where \(h_K(u)=\max\{u.x: x\in K\}\), \(h_K*\nu\) stands for the spherical convolution of the support function by a unique distribution \(\nu\in C^{-\infty}_0(\mathbb S^{n-1})\), and \(\Psi\) is uniformly continuous if and only \(\nu\) is a signed Borel measure (Theorem 1.1). Then author improves the latter theorem, by stating that \(\Psi\) is a Lipschitz continuous map such that its associated Lipschitz constant is bounded, up to a multiplicative constant, by the mean width of \(\Psi(\mathbb B^n)\), where \(\mathbb B^n\) stands for the unit ball in \(\mathbb R^n\). Next, by defining a weakly monotone Minkowski endomorphism, i.e., \(\Psi\) is monotone such that \(s(K)=\displaystyle\frac{1}{|\mathbb B^n|}\int_{\mathbb S^{n-1}}h_K(u)udu\) exists for all \(K\in\mathcal{K}^n\), the author states that a Minkowski endomorphism on \(\mathcal{K}^n\) for \(n\geq 3\), cannot be weakly monotone (Theorem 1.4). More general, the author states that there exist non-monotone even Minkowski endomorphisms (Theorem 3.5). Results on Minkowski valuations are also treated in the fourth section, see Corollary 1.8 and Theorem 1.9. The article is endowed by a helpful background material, see the second section.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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