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What is the average distance between two points on the sphere of radius n in a normed space? There is a natural choice of measure making this average-distance statistic into an affine invariant, and we explore the conjecture that the ℓ2 and ℓ∞ norms provide the two extreme values of this invariant on 
for every d.
2000 AMS Subject Classification:
We thank the referees for their careful reading and numerous helpful suggestions.
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