• Corpus ID: 232307758

On a conjectural symmetric version of Ehrhard's inequality

  title={On a conjectural symmetric version of Ehrhard's inequality},
  author={Galyna V. Livshyts},
As one of the results of this manuscript, we show that for any logconcave measure μ on R, any pair of symmetric convex sets K and L, and any λ ∈ [0, 1], μ((1− λ)K + λL)n ≥ (1− λ)μ(K)n + λμ(L)n , where cn ≥ n−4−o(1). This constitutes progress towards the Dimensional BrunnMinkowski conjecture (see Gardner, Zvavitch [45], Colesanti, L, Marsiglietti [27]). Moreover, our bound improves for various special classes of log-concave measures. However, most of the paper is dedicated to studying sharp… 
4 Citations
Sharp Estimates for the Gaussian Torsional Rigidity with Robin Boundary Conditions
In this paper we provide a comparison result between the solutions to the torsion problem for the Hermite operator with Robin boundary conditions and the one of a suitable symmetrized problem.
A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures
We show that for any log-concave measure μ on R, any pair of symmetric convex sets K and L, and any λ ∈ [0, 1], μ((1 − λ)K + λL) ≥ (1 − λ)μ(K) + λμ(L) , where cn ≥ n. This constitutes progress
Reverse Alexandrov-Fenchel inequalities for zonoids
The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume V (K1,K2,K3, . . . ,Kn) of convex bodies K1, . . . ,Kn in R by the product of the mixed volumes V (K1,K1,K3, . . .


Gaussian mixtures: entropy and geometric inequalities
A correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure is obtained and sharp constants in Khintchine inequalities for vectors uniformly distributed on the unit balls with respect to $p-norms are derived.
Sharp Poincaré-Type Inequality for the Gaussian Measure on the Boundary of Convex Sets
A sharp Poincare-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian
Thin shell implies spectral gap via a stochastic localization scheme
  • Geom. Funct. Anal., April 2013, Volume 23, Issue 2, pp 532-569
  • 2013
Isoperimetric problems for convex bodies and a localization lemma
This lemma is a general “Localization Lemma” that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable and is illustrated by showing how a number of well-known results can be proved using it.
Elliptic Partial Diferential Equations of Second Order
  • Springer- Verlag, Berlin, Heidelberg, New York,
  • 1977
On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
We extend the Prekopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prekopa—Leindler and Brunn-Minkowski theorems by introducing the notion of
The Brunn-Minkowski inequality in Gauss space
Symmetry of minimizers of a Gaussian isoperimetric problem
We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. The second term is in competition with the perimeter,
Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities
Given a probability measure $$\mu $$μ supported on a convex subset $$\Omega $$Ω of Euclidean space $$(\mathbb {R}^d,g_0)$$(Rd,g0), we are interested in obtaining Poincaré and log-Sobolev type
Asymptotic Geometric Analysis
Convex bodies: Classical geometric inequalities Classical positions of convex bodies Isomorphic isoperimetric inequalities and concentration of measure Metric entropy and covering numbers estimates