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

By Emanuel Milman, Liran Rotem. 2013

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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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/192ca91179120388afb8409adc92817937739fd2

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In Applied Mathematics

Citations in the corpus

Sharp isoperimetric inequalities via the ABP method (2013, unknown): “More recently, after posting the preprint version of the present paper, Milman and Rotem [39] have found an alternative proof of our isoperimetric inequalitiy, Theorem 1.3 above ([39] mentions that the same has been found independenly by Nguyen).”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, background): “[58] E. Milman and L. Rotem, Complemented Brunn-Minkowski Inequalities and Isoperimetry for Homogeneous and Non- Homogeneous Measures, Advances in Math., 262, 867–908, (2014).”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, result): “1 Let us mention that the usage of Brunn-Minkowski inequalities is a powerful tool to prove isoperimetric and other related geometric inequalities; for recent results in this direction we refer to Kolesnikov and Milman [35], and Milman and Rotem [47].”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, background): “4, this is actually known to hold for arbitrary Borel sets K (see [15, 54]).”
(K, N)-Convexity and the Curvature-Dimension Condition for Negative N (2013, background): “My gratitude also goes to Frank Morgan for drawing my attention to [25] and [18], and to Emanuel Milman for his helpful comments on the background of [25] and [18].”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “Let us also mention that the isoperimetric problem has already been studied in presence of (mild) singularities of the space: mostly for conical manifolds [49, 53] and polytopes [52].”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, background): “However, on a general weighted Riemannian manifold , our constant Note that in the Euclidean case, the CD(0 , N ) condition with N ∈ R corresponds to Borell’s class of convex measures [5], also known as “1 /N -concave measures” (cf. [34]).”
Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, background): “…measures μ on ( R n , |·| ) satisfying the CD(0 , N ) condition with 1 N ∈ [ −∞ , 1 /n ] have already been studied by Borell [12] (cf. [14]), who dubbed them convex (or 1 N -concave) measures; see [9, 11, 31, 49, 53] for a more detailed account of these measures and their useful properties.”