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- W2952318390 abstract "We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures $mu$, of the classical dual Sudakov Minoration on the expectation of the supremum of a Gaussian process: begin{equation} label{eq:abstract} M(Z_p(mu), C int ||x||_K dmu cdot K) leq exp(C p) ;;, forall p geq 1 . end{equation} Here $K$ is an origin-symmetric convex body, $Z_p(mu)$ is the $L_p$-centroid body associated to $mu$, $M(A,B)$ is the packing-number of $B$ in $A$, and $C > 0$ is a universal constant. The Program consists of first establishing 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 (ref{eq:abstract}) for all $p geq n$ and provide a new direct proof of Sudakov Minoration via The Program." @default.
- W2952318390 created "2019-06-27" @default.
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- W2952318390 date "2016-10-28" @default.
- W2952318390 modified "2023-09-23" @default.
- W2952318390 title "Generalized Dual Sudakov Minoration via Dimension Reduction - A Program" @default.
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