323 research outputs found
Sharpening Geometric Inequalities using Computable Symmetry Measures
Many classical geometric inequalities on functionals of convex bodies depend
on the dimension of the ambient space. We show that this dimension dependence
may often be replaced (totally or partially) by different symmetry measures of
the convex body. Since these coefficients are bounded by the dimension but
possibly smaller, our inequalities sharpen the original ones. Since they can
often be computed efficiently, the improved bounds may also be used to obtain
better bounds in approximation algorithms.Comment: This is a preprint. The proper publication in final form is available
at journals.cambridge.org, DOI 10.1112/S002557931400029
Quantitative anisotropic isoperimetric and Brunn-Minkowski inequalities for convex sets with improved defect estimates
In this paper we revisit the anisotropic isoperimetric and the
Brunn-Minkowski inequalities for convex sets. The best known constant
depending on the space dimension in both inequalities is due
to Segal [\ref{bib:Seg.}]. We improve that constant to for convex sets
and to for centrally symmetric convex sets. We also conjecture, that the
best constant in both inequalities must be of the form i.e., quadratic
in The tools are the Brenier's mapping from the theory of mass
transportation combined with new sharp geometric-arithmetic mean and some
algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.Comment: Recently Emanuel Milman from Technion pointed out a mistake, where I
missed an n^{1/2} on the right hand side of (3.18). The correction of the
mistake leads to a reproduction of the already known (Legal [35]) constant
Cn^
Stability results for some fully nonlinear eigenvalue estimates
In this paper, we give some stability estimates for the Faber-Krahn
inequality relative to the eigenvalues of Hessian operatorsComment: 18 pages, the proofs of Lemma 4.3 and Theorem 4.1 were clarifie
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