Realization of configurations and the Löwner ellipsoid (Q5951095)

Recall the following theorem [cf. \textit{K.~Leichtweiß}, Konvexe Mengen, Springer-Verlag, Berlin--Heidelberg--New York (1980; Zbl 0442.52001), Lemma~13.2]: John's theorem. Any compact convex body in \({\mathbb R}^m\) is contained in a~unique filled ellipsoid of minimum volume. It is proved that for a~regular simplex such an ellipsoid is the circumscribed ball. It is also proved that if \(A\subset S^{m-1}:=\bigl\{x\in{\mathbb R}^m:\|x\|=1\bigr\}\) is closed and \(\mu(A)\geq{l\over m+1}\mu(S^{m-1})\) for some positive integer~\(l\leq m\), then there exists a~regular simplex inscribed in \(S^{m-1}\) such that the vertices of one of its \(l\)--faces lie in~\(A\). Using these results the main result of the paper is obtained. It reads as follows: Theorem. If an ellipsoid~\(E\) and a~ball \(B\subset{\mathbb R}^m\) are such that the exterior of~\(E\) contains no larger than \(1/(m+1)\) fraction (in measure) of the sphere bounding~\(B\), then \text{vol}\((E)\geq \)\text{vol}\((B)\). The problem of the existence of a~pair of points with a~given spherical distance belonging to the set of positive measure is considered.