Higher-order Lp isoperimetric and Sobolev inequalities

Let ‖·‖ be the Euclidean norm on Rn and γn the (standard) Gaussian measure on Rn with density (2π)−n/2e. It is proved that there is a numerical constant c>0 with the following property: if K is an arbitrary convex body in Rn with γn(K)≥1/2, then to each sequence u1,…,um∈Rn with ‖u1‖,…,‖um‖≤c there correspond signs e1,…,em=±1 such that ∑mi=1eiui∈K. This improves the well-known result obtained by Spencer [Trans. Amer. Math. Soc.289, 679–705 (1985)] for the n-dimensional cube. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 351–360, 1998

Estimates for measures of lower dimensional sections of convex bodies

Volume inequalities of convex bodies from cosine transforms on Grassmann manifolds

Here we extend a result by J. Bourgain, J. Lindenstrauss, V.D. Milman on the number of random Minkowski symmetrizations needed to obtain an approximated ball, if we start from an arbitrary convex body in ℝn. We also show that the number of “deterministic” symmetrizations needed to approximate an Euclidean ball may be significantly smaller than the number of “random” ones.KeywordsConvex HullConvex BodyNumerical ConstantEuclidean BallDual NormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A Riesz representation theorem for functionals on log-concave functions

The paper discusses the existence of a continuous extension of functions that are defined on subsets of ℝ n and whose values are convex bodies in ℝ n . This problem arose in convex geometry in connection with the notion, recently introduced in algebraic geometry, of convex Newton-Okounkov bodies.

The geometry of smooth convex hypersurfaces in euclidean n-space Rn on the one hand and the geometry of the boundary of arbitrary convex bodies in Rn on the other are closely related (see [1] §17). The former belongs to differential geometry, the latter to geometric convexity. Some theorems have a differential geometric version as well as a convexity version; these depend on each other.

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

Sobolev and isoperimetric inequalities with monomial weights

Affine vs. Euclidean isoperimetric inequalities

In this paper, we will denote by convex figure a compact convex subset of the n-dimensional Euclidean space ℝn, and by convex body a convex figure with non-empty interior. The principal kinematic formula in integral geometry gives the measure of the set of congruent convex bodies intersecting with a fixed convex body. Specifically, let K, L be two convex bodies in ℝn and G(n) the group of special motions in ℝn. Each element, g: ℝn → ℝn, of G(n) can be represented bywhere b∈ℝn and e is an orthogonal matrix of determinant 1. Let μ be the Haar measure on G(n) normalized as follows: Let μ:ℝn × SO(n) → G(n) be defined by φ(t, e)x = ex + t, xeℝn, where SO(n) is the rotation group of ℝn. If v is the unique invariant probability measure on SO(n), η is the Lebesgue measure on ℝn, then μ is chosen as the pull back measure of η⊗v under φ−1. If Wi(K), Wi(L) are the quermassintegrals of K, L, i= 0, 1,…, n, the principal kinematic formula states thatwhere ωn is the volume of the unit n–ball.

Chapter 4 Convex geometry and functional analysis

By using optimal mass transport theory we prove a sharp isoperimetric inequality in $${\textsf {CD}} (0,N)$$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of n-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of Brendle (Comm Pure Appl Math 2021:13717, 2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

For a convex body K C iR, the kth projection function of K assigns to any k-dimensional linear subspace of RI the k-volume of the or thogonal projection of K to that subspace. Let K and Ko be convex bodies in Rn, and let Ko be centrally symmetric and satisfy a weak regularity assump tion. Let i, j E N be such that 1 < i < j < n-2 with (i, j) :A (1, n-2). Assume that K and Ko have proportional ith projection functions and proportional jth projection functions. Then we show that K and Ko are homothetic. In the particular case where Ko is a Euclidean ball, we thus obtain characteri zations of Euclidean balls as convex bodies having constant i-brightness and constant j-brightness. This special case solves Nakajima's problem in arbitrary dimensions and for general convex bodies for most indices (i, j).

The minimal Orlicz surface area