Geodesic
Realization of Configurations and the Loewner Ellipsoid
Matematičeskie zametki, Tome 69 (2001) no. 2, pp. 171-180

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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. 
@article{MZM_2001_69_2_a1,
     author = {S. A. Bogatyi},
     title = {Realization of {Configurations} and the {Loewner} {Ellipsoid}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {171--180},
     publisher = {mathdoc},
     volume = {69},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_69_2_a1/}
}
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S. A. Bogatyi. Realization of Configurations and the Loewner Ellipsoid. Matematičeskie zametki, Tome 69 (2001) no. 2, pp. 171-180. http://geodesic.mathdoc.fr/item/MZM_2001_69_2_a1/