Emanuel Milman

Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures

Abstract Elementary proofs of sharp isoperimetric inequalities on a normed space ( R n , ‖ ⋅ ‖ ) equipped with a measure μ = w ( x ) d x so that w p is homogeneous are provided, along with a characterization of the corresponding equality cases. When p ∈ ( 0 , ∞ ] and in addition w p is assumed concave, the result is an immediate corollary of the Borell–Brascamp–Lieb extension of the classical Brunn–Minkowski inequality, providing a new elementary proof of a recent Cabre–Ros-Oton–Serra result. When p ∈ ( − 1 / n , 0 ) , the relevant property turns out to be a novel “q-complemented Brunn–Minkowski” inequality: ∀ λ ∈ ( 0 , 1 ) ∀ Borel sets A , B ⊂ R n such that μ ( R n ∖ A ) , μ ( R n ∖ B ) ∞ , μ ⁎ ( R n ∖ ( λ A + ( 1 − λ ) B ) ) ≤ ( λ μ ( R n ∖ A ) q + ( 1 − λ ) μ ( R n ∖ B ) q ) 1 / q , which we show is always satisfied by μ when w p is homogeneous with 1 q = 1 p + n ; in particular, this is satisfied by the Lebesgue measure with q = 1 / n . This gives rise to a new class of measures, which are “complemented” analogues of the class of convex measures introduced by Borell, but which have vastly different properties. The resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of Canete–Rosales and Howe. The isoperimetric and Brunn–Minkowski type inequalities also extend to the non-homogeneous setting, under a certain log-convexity assumption on the density. Finally, we obtain functional, Sobolev and Nash-type versions of the studied inequalities.

On the Mean-Width of Isotropic Convex Bodies and their Associated Lp-Centroid Bodies

For any origin-symmetric convex body

Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems

K

Approachable sets of vector payoffs in stochastic games

in

Inner Regularization of Log-Concave Measures and Small-Ball Estimates

\mathbb{R}^n

A Comment on the Low-Dimensional Busemann–Petty Problem

in isotropic position, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \] where

Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities

M^*(K)

M -Estimates for Isotropic Convex Bodies and Their L q -Centroid Bodies

denotes (half) the mean-width of

Sharp Poincaré-Type Inequality for the Gaussian Measure on the Boundary of Convex Sets

K

Isoperimetric inequalities on weighted manifolds with boundary

,