Localization Technique on the Sphere and the Gromov–Milman Theorem on the Concentration Phenomenon on Uniformly Convex Sphere

We give a simpler proof of the Gromov–Milman theorem on concentration phenomenon on uniformly convex sphere. We also outline Rohlin’s theory of measurable partitions used in the proof. The purpose of this note is to present a localization technique for the sphere S on an example of the Gromov–Milman theorem [Gr-M] about the concentration phenomenon on uniformly convex spheres. This result was obtained in [Gr-M] in a some more general setting. Our approach follows the same general reasoning, but is simpler and more direct than the original approach. We also outline Rohlin’s theory of measurable partitions, which is used in the proof. Note that the terminology of “localization” was introduced for R by L. Lovász and M. Simonovits [L-S1, L-S2]. [Gr-M] did not use such terminology and also did not put the scheme of localization explicitly. Note. K. Ball has informed us recently that he, jointly with R. Villa, found an extremely short proof of the Gromov–Milman theorem for uniformly convex sphere as an application of the Prekopa–Leindler inequality (see, e.g., [P]). 1. Related Definitions and Formulation of the Gromov–Milman Theorem Definition 1.1. Let us say that a finite dimensional normed space X = (R, ‖·‖) has modulus of convexity at least δ(ε) > 0 for ε > 0, if for all vectors x, y ∈ X such that ‖x‖ = ‖y‖ = 1 and ‖x− y‖ ≥ ε we have ‖ 2 ‖ ≤ 1− δ(ε). We may assume δ(ε) to be a monotone increasing function of positive ε. Denote by K(X) := {x ∈ X : ‖x‖ ≤ 1} the unit ball of X and by S(X) := {x ∈ X : ‖x‖ = 1} the unit sphere of X. Work partially supported by a BSF Grant.