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An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture
We prove an almost constant lower bound of the isoperimetric coefficient in the KLS
conjecture. The lower bound has the dimension dependency d^-o_d (1) d-od (1). When the …
conjecture. The lower bound has the dimension dependency d^-o_d (1) d-od (1). When the …
Orlicz projection bodies
Minkowski's projection bodies have evolved into Lp projection bodies and their asymmetric
analogs. These all turn out to be part of a far larger class of Orlicz projection bodies. The …
analogs. These all turn out to be part of a far larger class of Orlicz projection bodies. The …
On the role of convexity in isoperimetry, spectral gap and concentration
We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality,
spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions …
spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions …
[كتاب][B] Local 𝐿^{𝑝}-Brunn–Minkowski inequalities for 𝑝< 1
A Kolesnikov, E Milman - 2022 - ams.org
Abstract The $ L^ p $-Brunn–Minkowski theory for $ p\geq 1$, proposed by Firey and
developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its $ L …
developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its $ L …
Robust covariance estimation under norm equivalence
S Mendelson, N Zhivotovskiy - 2020 - projecteuclid.org
Let X be a centered random vector taking values in R^d and let Σ=E(X⊗X) be its covariance
matrix. We show that if X satisfies an L_4-L_2 norm equivalence (sometimes referred to as …
matrix. We show that if X satisfies an L_4-L_2 norm equivalence (sometimes referred to as …
Relative entropy of cone measures and centroid bodies
Let $ K $ be a convex body in $\mathbb R^ n $. We introduce a new affine invariant, which
we call $\Omega_K $, that can be found in three different ways: as a limit of normalized …
we call $\Omega_K $, that can be found in three different ways: as a limit of normalized …
Sparsifying sums of norms
For any norms N_1,...,N_m on R^n and N(x):=N_1(x)+⋯+N_m(x), we show there is a
sparsified norm ̃N(x)=w_1N_1(x)+⋯+w_mN_m(x) such that |N(x)-̃N(x)|\leqslantεN(x) for all …
sparsified norm ̃N(x)=w_1N_1(x)+⋯+w_mN_m(x) such that |N(x)-̃N(x)|\leqslantεN(x) for all …
Efficient certificates of anti-concentration beyond gaussians
A set of high dimensional points X ={x_1,x_2,...,x_n\}⊆R^d in isotropic position is said to be
δ-anti concentrated if for every direction v, the fraction of points in X satisfying …
δ-anti concentrated if for every direction v, the fraction of points in X satisfying …
Centroid bodies and the logarithmic Laplace transform–a unified approach
We unify and slightly improve several bounds on the isotropic constant of high-dimensional
convex bodies; in particular, a linear dependence on the bodyʼs ψ2 constant is obtained …
convex bodies; in particular, a linear dependence on the bodyʼs ψ2 constant is obtained …
A characterization of dimension free concentration in terms of transportation inequalities
N Gozlan - 2009 - projecteuclid.org
The aim of this paper is to give a characterization of the dimension free concentration of
measure phenomenon in terms of transportation-cost inequalities. We apply this theorem to …
measure phenomenon in terms of transportation-cost inequalities. We apply this theorem to …