A characterization of the \(MM^*\)-position of a convex body in terms of covariance matrices
In 2000, A. Giannopoulos and V. Milman asked a question concerning positions of a convex body \(K\) which minimize \(\{M(TK) M^*(TK): T \in GL(n) \}\), where \[ M(K):={1 \over n\kappa_n} \int_{S^{n-1}} \| x\| _K d \sigma(x), \] with \(\kappa_n\) being the \(n\)-volume of the unit Euclidean ball and \(\| \cdot\| _K\) the gauge of \(K\); \[ M^*(K):=M(K^\circ), \] with \(K^\circ\) being the polar of \(K\). They found a necessary condition for a centrally symmetric convex body \(K\) to minimize \(\{M(TK) M^*(TK): T \in GL(n) \}\). The main goal of the paper under review is to show that conditions of this kind are also sufficient (Corollary 2.2 and Theorem 1.1). Moreover, since \[ M(K)={1 \over n \kappa_n} \int_{S^{n-1}}\varrho_K^{-1}d \sigma(u)= {1 \over \kappa_n} \tilde W_{n+1}(K), \] where \(\tilde W_i\) is the dual quermassintegral of order \(n-i\) (in the sense of Lutwak), the problem mentioned above is a particular case of that concerning characterizations of \(K\) which minimize \(\{\tilde W_i(TK) \tilde W_i((TK)^\circ): T \in GL(n)\}.\) Theorem 2.1 gives a solution of this general problem for \(i \in (-\infty, 0) \cup [n+1, \infty)\) and convex bodies ``smooth enough'' (i.e. with support functions of the body and its polar being twice continuously differentiable).
scientific article; zbMATH DE number 2114646