12 research outputs found
Bounding the norm of a log-concave vector via thin-shell estimates
Chaining techniques show that if X is an isotropic log-concave random vector
in R^n and Gamma is a standard Gaussian vector then E |X| < C n^{1/4} E |Gamma|
for any norm |*|, where C is a universal constant. Using a completely different
argument we establish a similar inequality relying on the thin-shell constant
sigma_n = sup ((var|X|^){1/2} ; X isotropic and log-concave on R^n).
In particular, we show that if the thin-shell conjecture sigma_n = O(1)
holds, then n^{1/4} can be replaced by log (n) in the inequality.
As a consequence, we obtain certain bounds for the mean-width, the dual
mean-width and the isotropic constant of an isotropic convex body.
In particular, we give an alternative proof of the fact that a positive
answer to the thin-shell conjecture implies a positive answer to the slicing
problem, up to a logarithmic factor.Comment: preliminary version, 13 page
Optimal Concentration of Information Content For Log-Concave Densities
An elementary proof is provided of sharp bounds for the varentropy of random
vectors with log-concave densities, as well as for deviations of the
information content from its mean. These bounds significantly improve on the
bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543,
2011).Comment: 15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with
more general sufficient conditions for equality in Theorem 2.3. Also some
minor corrections and added reference
Resonance-free Region in scattering by a strictly convex obstacle
We prove the existence of a resonance free region in scattering by a strictly
convex obstacle with the Robin boundary condition. More precisely, we show that
the scattering resonances lie below a cubic curve which is the same as in the
case of the Neumann boundary condition. This generalizes earlier results on
cubic poles free regions obtained for the Dirichlet boundary condition.Comment: 29 pages, 2 figure