The Sprawl Conjecture for Convex Bodies

Abstract

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:

• Primary 52A40
• Secondary 51F99
• 52B60

Keywords:

• affine isoperimetric problems
• curvature conditions
• convex geometry

Acknowledgments

We thank the referees for their careful reading and numerous helpful suggestions.