## Two notes on metric geometry

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- by Ralph Alexander
- Proc. Amer. Math. Soc.
**64**(1977), 317-320 - DOI: https://doi.org/10.1090/S0002-9939-1977-0442831-5
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## 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

## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**64**(1977), 317-320 - MSC: Primary 52A50
- DOI: https://doi.org/10.1090/S0002-9939-1977-0442831-5
- MathSciNet review: 0442831