Solidity indices for convex cones
A solidity index \(\sigma\) is a continuous real function defined on the set \(\Xi_n\) of non-trivial closed convex cones of \({\mathbb R}^n\), which is non-decreasing (with respect to set inclusion), invariant under orthogonal transformations, and so that it vanishes if the cone \(K\) is flat, whereas its value is \(1\) if \(K\) is a half-space. In this paper, the authors consider three particularly interesting solidity indices: the metric solidity index \(\sigma_{\operatorname{met}}(K)=\min_{P\in\Xi_n,\,P\mathrm{ is flat}} \delta(K\cap B_n,P\cap B_n)\) (here \(\delta\) is the Hausdorff distance and \(B_n\) the Euclidean unit ball); the volumetric solidity index \(\sigma_{\operatorname{vol}}(K)=2\operatorname{vol}(K\cap B_n)/\operatorname{vol}(B_n)\) (\(\operatorname{vol}\) denotes the \(n\)-dimensional volume); and the Frobenius solidity index \(\sigma_{\operatorname{frob}}(K) \) is the radius of the largest ball centered at a unit vector and contained in \(K\). They study properties of these indices (Lipschitz behavior of \(\sigma_{\operatorname{vol}}\) or numerical computation of \(\sigma_{\operatorname{frob}}\)) and establish all the relations between those three functions.
scientific article; zbMATH DE number 6121306