Projection bodies and valuations
The author considers maps \(Z:{\mathcal K}^n\to{\mathcal K}^n\) of the space \({\mathcal K}^n\) of convex bodies in \(\mathbb{R}^n\) into itself which are valuations, which means that \[ ZK_1+Z K_2=Z(K_1\cup K_2)+ Z(K_1\cap K_2) \] whenever \(K_1,K_2,K_1 \cup K_2\in{\mathcal K}^n\), and satisfy \[ Z(\Phi K)=|\det\Phi |\Phi^{-t} ZK\quad \text{and}\quad Z(K+x)= ZK \] for \(\Phi\in GL(n)\) and \(x\in\mathbb{R}^n\). The prototype of such a map is \(\Pi\), which associates with each \(K\in{\mathcal K}^n\) the projection body of \(K\). The author's main result asserts that if a map \(Z:{\mathcal P}^n\to{\mathcal P}^n\) \(({\mathcal P}^n\subset{\mathcal K}^n\) is the space of convex polytopes) satisfies the above conditions for all polytopes, then \(ZP=c\Pi P\) for all polytopes \(P\), where \(c\geq 0\) is a constant. Corollaries say that a map \(Z:{\mathcal K}^n\to{\mathcal K}^n\) with the above properties, which is either increasing (under set inclusion) or continuous (with respect to the Hausdorff metric), must be of the form \(Z=c\Pi\) with a constant \(c\geq 0\).
scientific article; zbMATH DE number 1888331
scientific article