On a multi-integral norm defined by weighted sums of log-concave random vectors

2022/08/12 by Skarmogiannis, Nikos · 1 citation
#52A40 #60D05 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Primary 46B06 #Secondary 52A23

paper · doi:10.48550/arxiv.2208.06365

Abstract

Let C and K be centrally symmetric convex bodies in \mathbb Rⁿ. We show that if C is isotropic then ‖\bf t‖_C^s,K=∫_C⋯∫_C‖∑_j=1^st_jx_j‖_K dx₁⋯ dx_s ≤ c₁L_C(log n)⁵ √(n)M(K)‖\bf t‖₂ for every s≥ 1 and \bf t=(t₁,… ,t_s)∈ \mathbb R^s, where L_C is the isotropic constant of C and M(K):=∫_S^n-1‖ξ‖_Kdσ(ξ). This reduces a question of V.~Milman to the problem of estimating from above the parameter M(K) of an isotropic convex body. The proof is based on an observation that combines results of Eldan, Lehec and Klartag on the slicing problem: If μ is an isotropic log-concave probability measure on \mathbb Rⁿ then, for any centrally symmetric convex body K in \mathbb Rⁿ we have that I₁(μ,K):=∫_\mathbb Rⁿ‖x‖_K dμ(x)≤ c₂√(n)(log n)⁵ M(K). We illustrate the use of this inequality with further applications.

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• Distances between non-symmetric convex bodies: optimal bounds up to polylog

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• Norms of weighted sums of log-concave random vectors
• Asymptotic shape of the convex hull of isotropic log-concave random vectors
• Estimates for measures of lower dimensional sections of convex bodies
• Pointwise Estimates for Marginals of Convex Bodies
• Random vectors in the isotropic position