Two notes on metric geometry

Alexander, Ralph

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

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.

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