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 $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets and to $Cn^5$ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form $Cn^2,$ i.e., quadratic in $n.$ 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