Related papers: Minkowski inequality for nearly spherical domains

We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement…

In this paper, we consider the following variational problem: \begin{eqnarray*} \inf_{u\in…

We prove limit equalities between the sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on an $m$-dimensional cube and ball and the corresponding constants for entire functions of exponential type.

A Minkowski symmetral of an $\alpha$-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given $\alpha$-concave function, there exists a sequence of Minkowski symmetrizations that…

In this paper, we prove a generalized Minkowski inequality holds for any smooth, $(k-1)$-convex, starshaped domain $\Omega.$ Our proof relies on the solvability of the degenerate $k$-Hessian equation on the exterior domain $\mathbb…

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability…

We give a systematic and thorough study of geometric notions and results connected to Minkowski's measure of symmetry and the extension of the well-known Minkowski functional to arbitrary, not necessarily symmetric convex bodies K on any…

The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a hypersurface in terms of the surface area, which is optimal on round spheres. In this paper we employ a locally constrained…

This paper explores the stability of Minkowski-type inequalities for hypersurfaces in warped product spaces. We establish a stability estimate that bounds the norm of the traceless second fundamental form of the hypersurface in terms of the…

Minkowski space is shown to be globally stable as a solution to the massive Einstein--Vlasov system. The proof is based on a harmonic gauge in which the equations reduce to a system of quasilinear wave equations for the metric, satisfying…

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq…

We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant…

In this note we derive a new Minkowski-type inequality for closed convex surfaces in the hyperbolic 3-space. The inequality is obtained by explicitly computing the area of the family of surfaces obtained from the normal flow and then…

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…

P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in $\mathbb{R}^3$. Following this work, we first…

In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} +…

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a…

This paper proves that for every convex body in R^n there exist 5n-4 Minkowski symmetrizations, which transform the body into an approximate Euclidean ball. This result complements the sharp c n log n upper estimate by J. Bourgain, J.…

This paper considers the radii functionals (circumradius, inradius, and diameter) as well as the Minkowski asymmetry for general (possibly non-symmetric) gauge bodies. A generalization of the concentricity inequality (which states that the…

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…