TY - JOUR

T1 - Generalized dual Sudakov minoration via dimension-reduction - A program

AU - Mendelson, Shahar

AU - Milman, Emanuel

AU - Paouris, Grigoris

N1 - Publisher Copyright:
© Instytut Matematyczny PAN, 2019.

PY - 2019

Y1 - 2019

N2 - We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures μ, of the classical dual Sudakov minoration on the expectation of the supremum of a Gaussian process: (Equation Presented) Here K is an origin-symmetric convex body, Zp(μ) is the Lp-centroid body associated to μ, M(A,B) is the packing number of B in A, and C > 0 is a universal constant. The program is to first establish a weak generalized dual Sudakov minoration, involving the dimension n of the ambient space, which is then self-improved to a dimension-free estimate after applying a dimension-reduction step. The latter step may be thought of as a conjectural "small-ball one-sided" variant of the Johnson{Lindenstrauss dimension-reduction lemma. We establish the weak generalized dual Sudakov minoration for a variety of log-concave probability measures and convex bodies (for instance, this step is fully resolved assuming a positive answer to the slicing problem). The separation dimension-reduction step is fully established for ellipsoids and, up to logarithmic factors in the dimension, for cubes, resulting in a corresponding generalized (regular) dual Sudakov minoration estimate for these bodies and arbitrary log-concave measures, which are shown to be (essentially) best possible. Along the way, we establish a regular version of (0.1) for all p ≥ n and provide a new direct proof of Sudakov minoration via the program.

AB - We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures μ, of the classical dual Sudakov minoration on the expectation of the supremum of a Gaussian process: (Equation Presented) Here K is an origin-symmetric convex body, Zp(μ) is the Lp-centroid body associated to μ, M(A,B) is the packing number of B in A, and C > 0 is a universal constant. The program is to first establish a weak generalized dual Sudakov minoration, involving the dimension n of the ambient space, which is then self-improved to a dimension-free estimate after applying a dimension-reduction step. The latter step may be thought of as a conjectural "small-ball one-sided" variant of the Johnson{Lindenstrauss dimension-reduction lemma. We establish the weak generalized dual Sudakov minoration for a variety of log-concave probability measures and convex bodies (for instance, this step is fully resolved assuming a positive answer to the slicing problem). The separation dimension-reduction step is fully established for ellipsoids and, up to logarithmic factors in the dimension, for cubes, resulting in a corresponding generalized (regular) dual Sudakov minoration estimate for these bodies and arbitrary log-concave measures, which are shown to be (essentially) best possible. Along the way, we establish a regular version of (0.1) for all p ≥ n and provide a new direct proof of Sudakov minoration via the program.

KW - Convex bodies

KW - Dimension-reduction

KW - Log-concave measures

KW - Packing and covering numbers

KW - Sudakov minoration

UR - http://www.scopus.com/inward/record.url?scp=85060847174&partnerID=8YFLogxK

U2 - 10.4064/sm170519-1-9

DO - 10.4064/sm170519-1-9

M3 - Article

SN - 0039-3223

VL - 244

SP - 159

EP - 202

JO - Studia Mathematica

JF - Studia Mathematica

IS - 2

ER -