Looking for selectors of star bodies
Following the line of previous papers, the author studies the possibilities of choosing ``good'' points inside a convex body or a star set. More precisely, let \({\mathcal F}\) be a family of star sets in \(\mathbb{R}^n\), a function \(s:{\mathcal F}\to \mathbb{R}^n\) is a selector for \({\mathcal F}\) if \(s(A)\in \text{ker} A\), where \(\text{ker} A\) denotes the set of \(a\in A\) such that \(A\) is star-shaped at \(a\). A family of selectors for the class \({\mathcal K}\) of convex bodies is then defined by means of concave and strictly increasing functions \(\varphi: [0, \infty) \to[0,\infty)\) as follows. For every star set \(A\) with radial function \(\rho_A\) let \(\Phi_A(x)\) be the integral on the unit sphere of the function \(\varphi\circ \rho_{A-x}\), where \(x\in \text{ker} A\). The author shows that if \(A\in {\mathcal K}\) then \(\Phi_A\) has a unique maximizer, called the radial centre of \(A\) induced by \(\varphi\). This centre is then extended to a selector for a wide class of star bodies. These selectors are proved to be equivariant under isometries and continuous with respect to a new metric, called star metric, which is topologically stronger then the Hausdorff metric.
scientific article; zbMATH DE number 1546666