Upper bound for the Dvoretzky dimension in Milman-Schechtman theorem

2016/12/12 by Huang, Han, Wei, Feng
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1612.03572

Abstract

For a symmetric convex body K⊂ℝⁿ, the Dvoretzky dimension k(K) is the largest dimension for which a random central section of K is almost spherical. A Dvoretzky-type theorem proved by V.~D.~Milman in 1971 provides a lower bound for k(K) in terms of the average M(K) and the maximum b(K) of the norm generated by K over the Euclidean unit sphere. Later, V.~D.~Milman and G. Schechtman obtained a matching upper bound for k(K) in the case when (M(K))/(b(K))>c((log(n))/(n))^(1)/(2). In this paper, we will give an elementary proof of the upper bound in Milman-Schechtman theorem which does not require any restriction on M(K) and b(K).

Related

• Radii of Euclidean sections of ℓ_p-balls
• Euclidean sections of convex bodies, series of lectures
• An isomorphic Dvoretzky-Milman Theorem using general random ensembles
• On a quantitative reversal of Alexandrov's inequality
• Superconcentration, and randomized Dvoretzky's theorem for spaces with\n 1-unconditional bases