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. 
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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/

[1] John F., “Extremum problems with inequalities as subsidiary conditions”, Studies and Essays, Presented to R. Courant on his 60th Birthday, New York, 1948, 187–204 | Zbl

[2] Zaguskin V. L., “Ob opisannykh i vpisannykh ellipsoidakh ekstremalnogo ob'ema”, UMN, 13:6 (1958), 89–92 | MR

[3] Gryunbaum B., Etyudy po kombinatornoi geometrii i teorii vypuklykh tel, Nauka, M., 1971 | Zbl

[4] Leikhtveis K., Vypuklye mnozhestva, Nauka, M., 1985

[5] Berzhe M., Geometriya, T. 2, Mir, M., 1984

[6] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987 | Zbl

[7] Larman D. G., Rogers C. A., “The realization of distances within sets in Euclidean space”, Mathematica, 19:1 (1972), 1–24 | MR | Zbl

[8] Leech J., “Some sphere packings in higher space”, Canad. J. Math., 16 (1964), 657–682 | MR | Zbl

[9] Leech J., “Notes on sphere packings”, Canad. J. Math., 19 (1967), 251–267 | MR | Zbl

[10] Conway J. H., “A group of order 8, 315, 553, 613, 086, 720, 000”, Bull. London Math. Soc., 1 (1969), 79–88 | DOI | MR | Zbl

[11] Boltyanskii V. G., Gokhberg I. Ts., Teoremy i zadachi kombinatornoi geometrii, Nauka, M., 1965 | Zbl

[12] Kahn J., Kalai G., “A counterexample to Borsuk's conjecture”, Bull. Amer. Math. Soc., 29:1 (1993), 60–62 | DOI | MR | Zbl

[13] Nilli A., “On Borsuk's problem”, Contemp. Math., 178 (1994), 209–210 | MR | Zbl

[14] Kalai G., “Combinatorics and convexity”, Proc. International Congress Math. Zürich, V. 1, Birkhäuser-Verlag, Basel, 1994, 1363–1374

[15] Raigorodskii A. M., “O razmernosti v probleme Borsuka”, UMN, 52:6 (1997), 181–182 | MR | Zbl

[16] Larman D. G., “A note on the realization of distances within sets in Euclidean space”, Comm. Math. Helv., 53 (1978), 529–535 | DOI | MR | Zbl

[17] Frankl P., Rödl V., “Forbidden intersections”, Trans. Amer. Math. Soc., 300:1 (1987), 259–286 | DOI | MR | Zbl

[18] Frankl P., Wilson R., “Intersection theorems with geometric consequences”, Combinatorica, 1 (1981), 357–368 | DOI | MR | Zbl

[19] Khadviger G., Debruner G., Kombinatornaya geometriya ploskosti, Nauka, M., 1965

[20] Steinhaus H., “Sur les distances des points des ensembles de mesure positive”, Fund. Math., 1 (1920), 99–104

[21] Kurepa S., “Note on the difference set of two measurable sets in $E^n$”, Glasnik Mat.-Fiz. Astronom. Ser. II, 15 (1960), 99–105 | MR | Zbl

[22] Kuczma M. E., Kuczma M., “An elementary proof and an extension of a theorem of Steinhaus”, Glasnik Mat., 6:1 (1971), 11–18 | MR | Zbl

[23] Sander W., “Verallgemeinerungen eines Satzes von H. Steinhaus”, Manuscripta Math., 18 (1976), 25–42 ; Errata, 20 (1977), 101–103 | DOI | MR | Zbl | DOI | MR | Zbl

[24] Sander W., “A generalization of a theorem of S. Piccard”, Proc. Amer. Math. Soc., 73:2 (1979), 281–282 | DOI | MR | Zbl

[25] Hall M. Jr., “On the sum and product of continued fractions”, Ann. of Math., 48 (1948), 966–993 | DOI

[26] Newhouse S., “Nondensity of axiom $A(a)$ on $S^2$”, Proc. Sympos. Pure Math., 14, AMS, Providence, R.I., 1970, 191–203 | MR

[27] Williams R. F., “How big is the intersection of two thick Cantor sets?”, Contemp. Math., 117 (1991), 163–175 | Zbl

[28] Raigorodskii A. M., “O khromaticheskom chisle prostranstva”, UMN, 55:2 (2000), 147–148 | MR | Zbl

[29] Raiskii D. E., “Realizatsiya vsekh rasstoyanii pri razbienii prostranstva $\mathbb R^n$ na $n+1$ chast”, Matem. zametki, 7:3 (1970), 319–323 | MR

[30] Hopf H., “Eine Verallgemeinerung bekannter Abbildungs und Überdeckungssätze”, Portugal. Math., 4 (1944), 129–139 | MR | Zbl

[31] Hadwiger H., “Eine Bemerkung zum Borsukschen Antipodensatz”, Vierteljschr. Naturf. Ges. Zürich, 89 (1944), 211–214 | MR | Zbl

[32] Larman D. G., “On the realization of distances within coverings of an $n$-sphere”, Mathematica, 14:2 (1967), 203–206 | MR | Zbl

[33] Bogatyi S. A., “Topologicheskie metody v kombinatornykh zadachakh”, UMN, 41:6 (1986), 37–48 | MR | Zbl

[34] Hadwiger H., “Überdeckung des Euklidischen Räumes durch kongruente Mengen”, Portugal. Math., 4 (1944), 238–242