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.