Two notes on metric geometry

Abstract

If μ , ∫ d μ = 1 \mu ,\smallint d\mu = 1 , is a signed Borel measure on the unit ball in E 3 {E^3} , it is shown that sup μ ∫ ∫ | p − q | d μ ( p ) d μ ( q ) = 2 {\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.