Brunn Minkowski Research Articles (Page 1)

Discrete Brunn–Minkowski inequality for subsets of the cube

Abstract This article describes two anisotropic area-preserving flows for plane curves, both of which are considered to deform one convex curve into another. Different monotonic entropy functions are identified under these flows, which can be utilized to derive two significant entropy inequalities: the log-Minkowski inequality and the curvature entropy inequality, as well as the Brunn–Minkowski inequality.

Abstract The infinitesimal forms of the $$L_{p}$$ L p -Brunn–Minkowski inequalities for variational functionals, such as the q-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for $$p \ge 0$$ p ≥ 0 . These formulations yield Poincaré-type inequalities related to these functionals. As an application, the $$L_{p}$$ L p -Brunn–Minkowski inequalities for torsional rigidity with $$0 \le p < 1$$ 0 ≤ p < 1 are confirmed for small smooth perturbations of the unit ball.

In 2003, associated with the radial Minkowski additions of star bodies, Zhao and Leng established the dual Brunn–Minkowski inequality for intersection bodies. In this paper, associated with the L_{p} -radial Minkowski combinations of star bodies, we firstly prove the L_{p} -dual Brunn–Minkowski inequality for intersection bodies. Further, associated with the L_{p} -Minkowski combinations of convex bodies, we give the L_{p} -Brunn–Minkowski inequality for star dualities of intersection bodies.

In the case of symmetries with respect to 𝑛 independent linear hyperplanes, a stability versions of the Logarithmic Brunn–Minkowski Inequality and the Logarithmic Minkowski Inequality for convex bodies are established.

The Dual Orlicz Brunn–Minkowski Inequality for the Polars of Mixed Projection Bodies

Companions to the Brunn–Minkowski inequality

$$L_p$$-Brunn–Minkowski inequality for projection bodies

We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a Riemannian surface for which a Brunn–Minkowski inequality holds with respect to a given volume form. In particular, we prove that under these assumptions, a family of constant-speed curves on a Riemannian surface satisfies the Brunn–Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature of its members is determined by a function κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} on the surface, and κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} satisfies the inequalityK+κ2-|∇κ|≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K + \\kappa ^2 - |\ abla \\kappa | \\ge 0 \\end{aligned}$$\\end{document}where K is the Gauss curvature.

We develop a calculus based on zonoids – a special class of convex bodies – for the expectation of functionals related to a random submanifold Z defined as the zero set of a smooth vector valued random field on a Riemannian manifold. We identify a convenient set of hypotheses on the random field under which we define its zonoid section, an assignment of a zonoid ζ(p) in the exterior algebra of the cotangent space at each point p of the manifold. We prove that the first intrinsic volume of ζ(p) is the Kac–Rice density of the expected volume of Z, while its center computes the expected current of integration over Z. We show that the intersection of random submanifolds corresponds to the wedge product of the zonoid sections and that the preimage corresponds to the pull-back.Combining this with the recently developed zonoid algebra, it allows to give a multiplication structure to the Kac–Rice formulas, resembling that of the cohomology ring of a manifold. Moreover, it establishes a connection with the theory of convex bodies and valuations, which includes deep results such as the Alexandrov–Fenchel inequality and the Brunn–Minkowski inequality. We export them to this context to prove two analogous new inequalities for random submanifolds. Applying our results in the context of Finsler geometry, we prove some new Crofton formulas for the length of curves and the Holmes–Thompson volumes of submanifolds in a Finsler manifold.

Abstract We show for of equal volume and that if , then (up to translation) is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by Figalli, van Hintum, and Tiba, the proof of which uses our current result. We additionally establish a similar sharp threshold for iterated sumsets.

The workshop convened researchers from algebraic geometry, convex geometry, and complex geometry to explore themes arising from the Alexandrov–Fenchel and Brunn–Minkowski inequalities. It featured three introductory talks delving into the basics of Lorentzian polynomials, valuations in convex geometry, and plurisubharmonic functions, that served as a foundation for the subsequent research talks. As anticipated, significant overlap emerged among the varied perspectives within these three areas, evident in the presentations and ensuing discussions.

While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in Rn. In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension n=1. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition (G,β) and nonempty sets A1,⋯,Am⊆R, equality holds iff for each S∈G, the set ∑i∈SAi is an interval. In the case of dimension n≥2 we will show that equality can hold if and only if the set ∑i=1mAi has measure 0.

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry.

In the setting of essentially non-branching metric measure spaces, we prove the equivalence between the curvature dimension condition CD(K, N ), in the sense of Lott-Sturm-Villani [Stu06a, Stu06b, LV09], and a newly introduced notion that we call strong Brunn-Minkowski inequality SBM(K, N ). This condition is a reinforcement of the generalized Brunn-Minkowski inequality BM(K, N ), which is known to hold in CD(K, N ) spaces. Our result is a first step towards providing a full equivalence between the CD(K, N ) condition and the validity of BM(K, N ), which has been recently proved in [MPR22] in the framework of weighted Riemannian manifolds.

Abstract We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature R i c N \mathrm {Ric}_N is bounded from below by a real number K K in every timelike direction satisfies the timelike curvature-dimension condition T C D q ( K , N ) \mathrm {TCD}_q(K,N) for all q ∈ ( 0 , 1 ) q\in (0,1) . The converse and a nonpositive-dimensional version ( N ≤ 0 N \le 0 ) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q q -Lorentz–Wasserstein distance as well as the characterization of q q -geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case.

Abstract The interplay between variational functionals and the Brunn–Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems à la Firey’s worn stone problem.

The curvature dimension condition CD(K,N), pioneered by Sturm and Lott–Villani in Sturm (2006a); Sturm (2006b); Lott and Villani (2009), is a synthetic notion of having curvature bounded below and dimension bounded above, in the non-smooth setting. This condition implies a suitable generalization of the Brunn–Minkowski inequality, denoted BM(K,N). In this paper, we address the converse implication in the setting of weighted Riemannian manifolds, proving that BM(K,N) is in fact equivalent to CD(K,N). Our result allows to characterize the curvature dimension condition without using neither the optimal transport nor the differential structure of the manifold.

In this paper, we explore questions regarding the Minkowski sum of the boundaries of convex sets. Motivated by a question suggested to us by V. Milman regarding the volume of [Formula: see text] where [Formula: see text] and [Formula: see text] are convex bodies, we prove sharp volumetric lower bounds for the Minkowski average of the boundaries of sets with connected boundary, as well as some related results.