Geometric means of ellipsoids and other convex bodies (Q2707501)

The aim of this paper is to define the so-called geometric mean of operators \(A_0\) and \(B_0\) on an inner-product (Hilbert) space \({\mathcal H}\). By recursion, the author defines operators \(A_{k+1}=(A_k+B_k)/2\) and \(B_{k +1}= 2(A_k^{-1}+ B_k^{-1})^{-1}\), where \(k\in\{0,1,2, \dots\}\), and next he shows that the two sequences \(\{A_k\}\) and \(\{B_k\}\) converge to the same limit, which is called the geometric mean of \(A_0\) and \(B_0\). If \(A\) is a positive definite operator on \({\mathcal H}\), then the set \(E_A=\{x: \langle Ax,x\rangle <1\}\) is called an ellipsoid. Thus the preceding definition leads also to the definition of the geometric mean of ellipsoids. Applying the support function in a similar way, the author also defines the geometric mean of convex bodies in a finite dimensional space.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00038].