Approximation of convex compact sets by ellipsoids. Ellipsoids of best approximation (Q2378013)

Let \(W\) be a convex compact set in \(\mathbb{R}^n\) and \(E(w,B)\) an ellipsoid described by its support function \(c(E,\psi)= (w,\psi)+ \sqrt{\psi^*B\psi}\), \(\psi\in\mathbb{R}^n\), where \(w\in\mathbb{R}^n\) is the center of the ellipsoid and \(B\) is a symmetric positive definite matrix of order \(n\). We assume that \(B= B^*\geq 0\), \(w\in W\) and the deviation of the ellipsoid \(E(w,B)\) from the compact set \(W\) is invariant under parallel translations of these sets and therefore it coincides with the deviation of the ellipsoid \(E(0,B)= E(w,B)- w\) centered at zero from the compact set \(W-w\). We define the deviation \(p\) of the ellipsoid \(E(w,B)\) from the convex compact set \(w\) as \(P(E,W)= \sqrt{\Phi(w,B)}\), where \[ \Phi(w,B)= \int_S [c^2(E(0,B),\psi)- c^2(W- w,\psi)]^2\,ds= \int_S [\psi^* B\psi-(c(W,\psi)- (w,\psi)^2]^2\,ds.\tag{1} \] The ellipsoid of best approximation for a convex compact set \(W\) in the sense of deviation \(p\) is an ellipsoid \(E(\widehat w,\widehat B)\) that minimizes the criterion (1): \(\Phi(\widehat w,\widehat B)=\min_w \min_{B= B^*\geq 0}\Phi(w,B)\). The author considers the problem of best approximation of a convex compact set in a finite-dimensional space by ellipsoid with respect to the special measure of deviation \(p\) of an ellipsoid from a compact set. An analytic description of ellipsoid of best approximation has also been given in the paper.