Brunn Minkowski Theory Research Articles

SL(n) contravariant function-valued valuations on polytopes

We present a complete classification of SL(n) contravariant, C(Rn∖{o})-valued valuations on polytopes, without any additional assumptions. It extends the previous results of the second author Li (2020) [10] which have a good connection with the Lp and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of SL(n) contravariant symmetric-tensor-valued valuations on polytopes.

Sharp affine weighted L 2 Sobolev inequalities on the upper half space

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.

The (Self-Similar, Variational) Rolling Stones

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.

Weighted Brunn-Minkowski Theory I: On weighted surface area measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in Rn – in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

TheLpchord Minkowski problem

AbstractChord measures are newly discovered translation-invariant geometric measures of convex bodies inRn{{\mathbb{R}}}^{n}, in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing theLp{L}_{p}chord measures is called theLp{L}_{p}chord Minkowski problem in theLp{L}_{p}Brunn-Minkowski theory, which includes theLp{L}_{p}Minkowski problem as a special case. This article solves theLp{L}_{p}chord Minkowski problem whenp>1p\gt 1and the symmetric case of0<p<10\lt p\lt 1.

On a shape derivative formula for star-shaped domains using Minkowski deformation

<abstract><p>We consider the shape derivative formula for a volume cost functional studied in previous papers where we used the Minkowski deformation and support functions in the convex setting. In this work, we extend it to some non-convex domains, namely the star-shaped ones. The formula happens to be also an extension of a well-known one in the geometric Brunn-Minkowski theory of convex bodies. At the end, we illustrate the formula by applying it to some model shape optimization problem.</p></abstract>

Curvature functionals on convex bodies

AbstractWe investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn–Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to f divergences of the cone measures of the convex body and its polar, namely the Kullback–Leibler divergence and the Rényi divergence.

Geometry of log-concave functions: the $$L_p$$ Asplund sum and the $$L_{p}$$ Minkowski problem

The aim of this paper is to develop a basic framework of the $$L_p$$ theory for the geometry of log-concave functions, which can be viewed as a functional “lifting” of the $$L_p$$ Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the $$L_p$$ Asplund sum of log-concave functions for all $$p>1$$ and the total mass, we obtain a Prékopa-Leindler type inequality and propose a definition for the first variation of the total mass in the $$L_p$$ setting. Based on these, we further establish an $$L_p$$ Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our $$L_p$$ surface area measure for log-concave functions. Consequently, the $$L_p$$ Minkowski problem for log-concave functions, which aims to characterize the $$L_p$$ surface area measure for log-concave functions, is introduced. The existence of solutions to the $$L_p$$ Minkowski problem for log-concave functions is obtained for $$p>1$$ under some mild conditions on the pre-given Borel measures.

The functional form of the dual mixed volume

This paper is to generalize the dual mixed volume of star bodies to that of dual quasi-concave functions by extending the radial Minkowski linear combination of star bodies to that of dual quasi-concave functions. We attempt to build up some functional versions of notions and inequalities from the dual Brunn-Minkowski theory. In particular, both the dual mixed Brunn-Minkowski inequality of star bodies and the dual Aleksandrov Fenchel inequality of star bodies are generalized to that of dual quasi-concave functions.

Real and complex hedgehogs, their symplectic area, curvature and evolutes

Classical (real) hedgehogs can be regarded as the geometrical realiza-tions of formal di¤erences of convex bodies in R n+1. Like convex bodies, hedgehogs can be identi…ed with their support functions. Adopting a pro-jective viewpoint, we prove that any holomorphic function h : C n ! C can be regarded as the 'support function' of a complex hedgehog H h in C n+1. In the same vein, we introduce the notion of evolute of such a hedgehog H h in C 2 , and a natural (but apparently hitherto unknown) notion of complex curvature, which allows us to interpret this evolute as the locus of the centers of complex curvature. It is of course permissible to think that the development of a 'Brunn-Minkowski theory for complex hedgehogs' (replacing Euclidean volumes by symplectic ones) might be a promising way of research. We give …rst two results in this direction. We next return to real hedgehogs in R 2n endowed with a linear complex structure. We introduce and study the notion of evolute of a hedgehog. We particularly focus our attention on R 4 endowed with a linear Kahler structure determined by the datum of a pure unit quaternion. In parallel, we study the symplectic area of the images of the oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function for such an image. Finally, we consider brie ‡y the convolution of hedgehogs, and the particular case of hedgehogs in R 4n regarded as a hyperkahler vector space.

Further inequalities and properties of p-inner parallel bodies

AbstractA. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.

Orlicz Brunn-Minkowski theory

In this note, we mainly introduce the Orlicz Brunn-Minkowski theory from three aspects:Orlicz projection bodies and Orlicz centroid bodies,Orlicz addition and related volume inequalities, and Orlicz Minkowski problems.

Shephard Type Problem for Lp-mixed Blaschke-Minkowski homomorphisms

In 2006, Schuster introduced the concept of Blaschke-Minkowski homo-morphism of convex bodies. In this paper, we introduce the Lp -mixed Blaschke-Minkowski homomorphism in Lp -Brunn-Minkowski theory. We then further study the Shephard type problem involving an affirmative answer and two negative answers for Lp -mixed Blaschke-Minkowski homomorphism.

Bergman kernel and oscillation theory of plurisubharmonic functions

Based on Harnack’s inequality and convex analysis we show that each plurisubharmonic function is locally BUO (bounded upper oscillation) with respect to polydiscs of finite type but not for arbitrary polydiscs. We also show that each function in the Lelong class is globally BUO with respect to all polydiscs. A dimension-free BUO estimate is obtained for the logarithm of the modulus of a complex polynomial. As an application we obtain an approximation formula for the Bergman kernel that preserves all directional Lelong numbers. For smooth plurisubharmonic functions we derive a new asymptotic identity for the Bergman kernel from Berndtsson’s complex Brunn–Minkowski theory, which also yields a slightly better version of the sharp Ohsawa–Takegoshi extension theorem in some special cases.

Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications

The notion of intersection body is introduced by Lutwak in 1988, it is one of important research contents and led to the studies of Busemann-Petty problem in the Brunn-Minkowski theory. Based on the properties of the intersection bodies, Schuster introduced the notion of radial Blaschke-Minkowski homomorphisms and proved a lot of related inequalities. In this paper, by applying the dual mixed volume theory and analytic inequalities, we first give a lower bound of the dual quermassintegrals for the mixed radial Blaschke-Minkowski homomorphisms. As its an application, we get a reverse form of the well-known Busemann intersection inequality. Further, a Brunn-Minkowski type inequality of the Lp radial Minkowski sum for the dual quermassintegrals of mixed radial Blaschke-Minkowski homomorphisms is established, and then the intersection body version of this Brunn-Minkowski type inequality is yielded. From this, we not only extend Schuster's related result but also obtain the Brunn-Minkowski type inequalities of Lp harmonic radial sum and Lp radial Blaschke sum, respectively.

Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

AbstractWe establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants.Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $\psi$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $\mu$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $\mu$ equals $\deV(K, \cdot)$, up to a constant. A major step in the analysis concerns discrete measures $\mu$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures $\mu$, we use an approximation argument. This approach is also applied to the case where $G$ is decreasing and $\psi$ is increasing, and hence augments our previous work. When the measures $\mu$ are even, solutions that are origin-symmetric convex bodies are also provided under some mild conditions on $G$ and $\psi$. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $\mu$ is discrete or even.

A Steiner formula in the Lp Brunn Minkowski theory

We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameter p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory as well as the dual quermassintegrals of the dual Brunn Minkowski theory as special cases.Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the s-th mixed Lp affine surface area of a Minkowski outer parallel body for any real parameters p and s is also given.

Orlicz Addition for Measures and an Optimization Problem for the -divergence

AbstractThis paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

Asymmetric Orlicz Radial Bodies

Based on the L p -harmonic radial combination, Li and Wang researched the asymmetric L p -harmonic radial bodies, which belong to the asymmetric L p -Brunn-Minkowski theory initiated by Ludwig, Haberl and Schuster. In this paper, combined with Orlicz radial combination, we introduce the asymmetric Orlicz radial bodies and research their properties. Further, we also establish some inequalities for this concept.