Related papers: The functional volume product under heat flow

The Wills functional $\mathcal{W}(K)$ of a convex body $K$, defined as the sum of its intrinsic volumes $\mathrm{V}_i(K)$, turns out to have many interesting applications and properties. In this paper we make profit of the fact that it can…

We show that the thermal subadditivity of entropy provides a common basis to derive a strong form of the bounded difference inequality and related results as well as more recent inequalities applicable to convex Lipschitz functions, random…

We prove stability estimates for the Bakry-Emery bound on Poincar\'e and logarithmic Sobolev constants of uniformly log-concave measures. In particular, we improve the quantitative bound in a result of De Philippis and Figalli asserting…

We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$ defined as follows: ${\mathcal F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body…

It is shown that each continuous even Minkowski valuation on convex bodies of degree $1 \leq i \leq n - 1$ intertwining rigid motions is obtained from convolution of the $i$th projection function with a unique spherical Crofton…

We show differentiability of a class of Geroch's volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz…

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and…

This paper deals with an analog of the Mahler volume product related to the ${\cal J}$ transform acting in the class of geometric convex functions ${\rm{Cvx}}_0({\mathbb R}^n)$. We provide asymptotically sharp bounds for the quantity…

Using the inner product formula of the canonical Hilbert space of fractional Brownian motion on an interval $[0,T]$ with Hurst parameter $H\in (0,1)$ given by Alazemi et al., we show the asymptotic expansion of the norm of…

Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…

We prove a contraction property of Fock type spaces $\mathcal{L}_{\alpha}^p$ of log-subharmonic functions in $\mathbb{R}^n$. To prove the result, we demonstrate a certain monotonic property of measures of the superlevel set of the function…

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set…

We prove a very general sharp inequality of the H\"older--Young--type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the…

This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional…

The Blaschke Santal\'o inequality and the $L_p$ affine isoperimetric inequalities are major inequalities in convex geometry and they have a wide range of applications. Functional versions of the Blaschke Santal\'o inequality have been…

We prove sharp estimates for Fourier transforms of indicator functions of bounded open sets in ${\mathbb R}^n$ with real analytic boundary, as well as nontrivial lattice point discrepancy results. Both will be derived from estimates on…

We revisit the problem of the overdamped (large friction) limit of the Brownian dynamics in an inhomogeneous medium characterized by a position-dependent friction coefficient and a multiplicative noise (local temperature) in one space…

We show how Beckner's montonicity result on Hamming cube easily implies the monotonicity of a flow introduced by Janson in Hausdorff--Young inequality

We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy…

We establish a sharp upper-bound for the first non-zero even eigenvalue (corresponding to an even eigenfunction) of the Hilbert-Brunn-Minkowski operator associated to a strongly convex $C^2$-smooth origin-symmetric convex body $K$ in…