Realization of Configurations and the Loewner Ellipsoid

It is proved that any subset of an $(m-1)$-dimensional sphere of volume larger than $l(m+1)$ of the volume of the entire sphere contains $l+1$ points forming a regular $l$-dimensional simplex. As a corollary, it is obtained that, if the exterior of a given $m$-dimensional filled ellipsoid contains no more than the $1/(m+1)$ fraction of some sphere, then the volume of the ellipsoid is no less than the volume of the corresponding ball. The existence of a pair of points a given spherical distance apart in a set of positive measure is examined.