Two notes on metric geometry

Alexander, Ralph

doi:10.1090/S0002-9939-1977-0442831-5

Two notes on metric geometry
HTML articles powered by AMS MathViewer

by Ralph Alexander PDF

Proc. Amer. Math. Soc. 64 (1977), 317-320 Request permission

Abstract:

If $\mu ,\smallint d\mu = 1$, is a signed Borel measure on the unit ball in ${E^3}$, it is shown that ${\sup _\mu }\smallint \smallint | {p - q} |d\mu (p)d\mu (q) = 2$ with no extremal measure existing. Also, a class of simplices which generalizes the notion of acute triangle is studied. The results are applied to prove inequalities for determinants of the Cayley-Menger type.

References

Ralph Alexander and Kenneth B. Stolarsky, Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc. 193 (1974), 1–31. MR 350629, DOI 10.1090/S0002-9947-1974-0350629-3
Ralph Alexander, Generalized sums of distances, Pacific J. Math. 56 (1975), no. 2, 297–304. MR 513964
Göran Björck, Distributions of positive mass, which maximize a certain generalized energy integral, Ark. Mat. 3 (1956), 255–269. MR 78470, DOI 10.1007/BF02589412
Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
I. J. Schoenberg, On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space, Ann. of Math. (2) 38 (1937), no. 4, 787–793. MR 1503370, DOI 10.2307/1968835