A note on the metric geometry of the unit ball

作者：Hinrichs Aicke; Nickolas Peter^*; Wolf Reinhard

来源：Mathematische Zeitschrift, 2011, 268(3-4): 887-896.

DOI：10.1007/s00209-010-0700-y

• Euclidean unit ball
• Compact metric space
• Signed Borel measure
• Spaces of Borel measures
• Metric geometry
• Distance geometry
• Geometric constant

Denote by B(n) the unit ball in the Euclidean space R(n) and define
M(B(n)) = sup integral(Bn)integral(Bn) parallel to x - y parallel to d mu(x)d mu(y),
where the supremum is taken over all finite signed Borel measures mu on B(n) of total mass 1. In this paper, the value of M(B(n)) is computed explicitly for all n, and it is shown that for n > 1 no measure exists that achieves the supremum defining M(B(n)). These results generalize the work of Alexander (Proc Am Math Soc 64: 317-320, 1977) on M(B(3)).

• 出版日期2011-8