Minkowski Integral Inequality Revisited

Borell's generalized Prékopa-Leindler inequality: A simple proof

We present a simple proof of Christer Borell's general inequality in the Brunn-Minkowski theory. We then discuss applications of Borell's inequality to the log-Brunn-Minkowski inequality of Böröczky, Lutwak, Yang and Zhang.

From the Brunn-Minkowski inequality to a class of Poincaré type inequalities

We present an argument which leads from the Brunn-Minkowski inequality to a Poincare' type inequality on the boundary of convex bodies with smooth boundary and positive Gauss curvature

Bohr's inequality revisited

We survey several significant results on the Bohr inequality and presented its generalizations in some new approaches. These are some Bohr type inequalities of Hilbert space operators related to the matrix order and the Jensen inequality. An eigenvalue extension of Bohr's inequality is discussed as well.

关于平面偶极对偶拟质量积分的 $L_p$ Brunn--Minkowski 不等式

On the $L_p$ Brunn--Minkowski inequality for planar dual quermassintegrals

本文确定了平面偶极对偶拟质量积分的 $L_p$ Brunn--Minkowski 不等式的锐利范围，证明了 $2<q<4$ 时的 Minkowski 不等式并给出反例，完成了等号分类。

The weak elliptic Harnack inequality revisited

In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincaré inequality, for any regular Dirichlet form without killing part on a measure metric space, by using the lemma of growth and the John-Nirenberg inequality. We secondly show several equivalent characterizations of the weak elliptic Harnack inequality for any (not necessarily regular) Dirichlet form. We thirdly present some consequences of the weak elliptic Harnack inequality.

Hardy's inequality for Hermite expansions revisited

In this article, we give a short proof of Hardy's inequality for Hermite expansions of functions in the classical Hardy spaces $H^p({\mathbb R^n})$, by using an atomic decomposition of the Hardy spaces associated with the Hermite operators. When the space dimension is $1$, we obtain a new estimate of Hardy's inequality for Hermite expansions in $H^p({\mathbb R})$ for the range $0<p<1.$

A letter: The log-Brunn-Minkowski inequality for complex bodies

In this short note we explain why the log-Brunn-Minkowski conjecture is correct for complex convex bodies. We do this by relating the conjecture to the notion of complex interpolation, and appealing to a general theorem by Cordero-Erausquin.

Quantitative stability for the Brunn-Minkowski inequality

We prove a quantitative stability result for the Brunn-Minkowski inequality: if $|A|=|B|=1$, $t \in [τ,1-τ]$ with $τ>0$, and $|tA+(1-t)B|^{1/n}\leq 1+δ$ for some small $δ$, then, up to a translation, both $A$ and $B$ are quantitatively close (in terms of $δ$) to a convex set $K$.

An extension of Milman's reverse Brunn-Minkowski inequality

The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset\R^n$ compact, $$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$ where $|\cdot|$ denotes the Lebesgue measure on $\R^n$. In 1986 V. Milman {\bf [Mil 1]} discovered that if $B_1$ and $B_2$ are balls there is always a relative position of $B_1$ and $B_2$ for which a perturbed inverse of $(1)$ holds. More precisely:\lq\lq{\sl There exists a constant $C>0$ such that for all $n\in\N$ and any balls $B_1,B_2\subset\R^n$ we can find a linear transformation $u\colon\R^n\to\R^n$ with $|{\rm det}(u)|=1$ and $$|u(B_1)+B_2|^{1/n…

Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequality

By studying $L^p$-combinations of strongly isomorphic polytopes, we prove the equivalence of the $L^p$-Brunn-Minkowski inequality conjectured by Böröczky, Lutwak, Yang and Zhang to the local version of the inequality studied by Colesanti, Livshyts, and Marsiglietti and by Kolesnikov and Milman, settling a conjecture of the latter authors. In addition, we prove the local inequality in dimension $2$, yielding a new proof of the $L^p$-Brunn-Minkowski inequality in the plane.

On Multiple $L_p$-curvilinear-Brunn-Minkowski inequalities

We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $L_p$ space for multiple power parameter $\barα=(α_1, \cdots, α_{n+1})$ when $p>0$. Based on this $L_{p,\barα}$-curvilinear summation for sets and concept of {\it compression} of sets, the $L_{p,\barα}$-curvilinear-Brunn-Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of $L_{p,\barα}$ Borell-Brascamp-Lieb inequality, as well as its normalized version, for functions …

Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality

In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability . Here we show that the classical Shannon's entropy power inequality has a counterpart for the free analogue of entropy . The free entropy (introduced recently by the second named author), consistently with Boltzmann's formula $S=k\log W$, was defined via volumes of matricial microstates. Proving the free entropy power inequality naturally becomes a geometric question. Restricting the Minkowski sum of two sets means to specify …

On the stability of Brunn-Minkowski type inequalities

Log-Brunn-Minkowski inequality was conjectured by Boröczky, Lutwak, Yang and Zhang \cite{BLYZ}, and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors \cite{LMNZ}, that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequality for log-concave measures, which in the case of Gaussian measure was conjectured by Gardner and Zvavitch \cite{GZ}. In this note, we obtain stability results for both Log-Brunn-Minkowsk…

量子KKL型不等式再探

Quantum KKL-type Inequalities Revisited

本文在量子框架下发展随机限制方法，建立量子Eldan-Gross不等式、量子Talagrand等周不等式及相关量子KKL型不等式，恢复并替代性地回答了量子KKL猜想。

Brunn-Minkowski inequalities in product metric measure spaces

Given one metric measure space $X$ satisfying a linear Brunn-Minkowski inequality, and a second one $Y$ satisfying a Brunn-Minkowski inequality with exponent $p\ge -1$, we prove that the product $X\times Y$ with the standard product distance and measure satisfies a Brunn-Minkowski inequality of order $1/(1+p^{-1})$ under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn-Minkowski inequality is obtained in $X\times Y$ when $Y$ satisfies a Prékopa-Leind…

The local logarithmic Brunn-Minkowski inequality for zonoids

The aim of this note is to show that the local form of the logarithmic Brunn-Minkowski conjecture holds for zonoids. The proof uses a variant of the Bochner method due to Shenfeld and the author.

Higher order Poincare inequalities and Minkowski-type inequalities

We observe some higher order Poincare-type inequalities on a closed manifold, which is inspired by Hurwitz's proof of the Wirtinger's inequality using Fourier theory. We then give some geometric implication of these inequalities by applying them on the sphere. More specifically, by applying them to the support function of a convex hypersurface in the Euclidean space, we obtain some sharp Minkowski-type inequalities, such as a stability inequality for the classical Minkowski inequality and the Alexandrov-Fenchel inequality.

Blaschke-Santaló inequalities for Minkowski and Asplund endomorphisms

It is shown that each monotone Minkowski endomorphism of convex bodies gives rise to an isoperimetric inequality which directly implies the classical Urysohn inequality. Among this large family of new inequalities, the only affine invariant one - the Blaschke-Santaló inequality - turns out to be the strongest one. A further extension of these inequalities to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover, for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequalities for log-concave functions is established which generalizes…

On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application

In this paper, we use a new method to obtain the necessary and sufficient condition guaranteeing the validity of the Minkowski-Hölder type inequality for the generalized upper Sugeno integral in the case of functions belonging to a wider class than the comonotone functions. As a by-product, we show that the Minkowski type inequality for seminormed fuzzy integral presented by Daraby and Ghadimi in General Minkowski type and related inequalities for seminormed fuzzy integrals, Sahand Communications in Mathematical Analysis 1 (2014) 9--20 is not true. Next, we study the Minkowski-Hölder inequalit…

关于 p-布朗-闵科夫斯基和布拉斯坎-利布不等式

On p-Brunn-Minkowski and Brascamp-Lieb inequalities

本文证明了对称 p-布朗-闵科夫斯基不等式的新充分条件，并在所有维度中建立了 L_q 球体的局部对数布朗-闵科夫斯基不等式。

The dimensional Brunn-Minkowski inequality in Gauss space

Let $γ_n$ be the standard Gaussian measure on $\mathbb{R}^n$. We prove that for every symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $λ\in(0,1)$, $$γ_n(λK+(1-λ)L)^{\frac{1}{n}} \geq λγ_n(K)^{\frac{1}{n}}+(1-λ)γ_n(L)^{\frac{1}{n}},$$ thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure. We also show that, for a fixed $λ\in(0,1)$, equality is attained if and only if $K=L$.

对偶均质积分的Brunn-Minkowski不等式与逆向等周不等式

Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals

本文在对偶Brunn-Minkowski理论中建立了两个新几何不等式：证明了Lutwak猜想的原点对称凸体对偶均质积分的Brunn-Minkowski不等式，并推广Ball体积比不等式得出在John位置下立方体最大化对偶均质积分的逆向等周不等式。

A note on an $L^p$-Brunn-Minkowski inequality for convex measures in the unconditional case

We consider a different $L^p$-Minkowski combination of compact sets in $\mathbb{R}^n$ than the one introduced by Firey and we prove an $L^p$-Brunn-Minkowski inequality, $p \in [0,1]$, for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function $t \mapsto μ(t^{\frac{1}{p}} A)$, $p \in (0,1]$, for unconditional convex measures $μ$ and unconditional convex body $A$ in $\mathbb{R}^n$. We also prove that the (B)-conjecture for all uniform measures is equivalent to the (B)…

非交换闵可夫斯基积分不等式与融合环的单位ary分类化准则

Noncommutative Minkowski integral inequality and a unitary categorification criterion for fusion rings

本文证明了非交换闵可夫斯基积分不等式，并给出了融合环的单位ary分类化准则。

Sharp quantitative stability of the planar Brunn-Minkowski inequality

We prove a sharp stability result for the Brunn-Minkowski inequality for $A,B\subset\mathbb{R}^2$. Assuming that the Brunn-Minkowski deficit $δ=|A+B|^{\frac{1}{2}}/(|A|^\frac12+|B|^\frac12)-1$ is sufficiently small in terms of $t=|A|^{\frac{1}{2}}/(|A|^{\frac{1}{2}}+|B|^{\frac{1}{2}})$, there exist homothetic convex sets $K_A \supset A$ and $K_B\supset B$ such that $\frac{|K_A\setminus A|}{|A|}+\frac{|K_B\setminus B|}{|B|} \le C t^{-\frac{1}{2}}δ^{\frac{1}{2}}$. The key ingredient is to show for every $ε>0$, if $δ$ is sufficiently small then $|co(A+B)\setminus (A+B)|\le (1+ε)(|co(A)\setminus A…

On the analogue of the concavity of entropy power in the Brunn-Minkowski theory

Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in {\it On the similarity of the entropy power inequality and the Brunn-Minkowski inequality} (IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the $\frac{1}{n}$-concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We investigate this conjecture and study its relationship with geometric inequalities.

Dual Orlicz-Brunn-Minkowski theory: Orlicz $φ$-radial addition, Orlicz $L_ϕ$-dual mixed volume and related inequalities

This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz $φ$-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz $φ$-radial addition of two star bodies, we derive a formula for the Orlicz $L_ϕ$-dual mixed volume. Moreover, a dual Orlicz-Minkowski inequality for the Orlicz $L_ϕ$-dual mixed volume, a dual Orlicz isoperimetric inequality for the Orlicz $L_ϕ$-dual surface area and a dual Orlicz-Urysohn inequality for the Orlicz $L_ϕ$-harmonic mean radius are…

Weak anisotropic Hardy inequality: essential self-adjointness of drift-diffusion operators on domains in $\mathbb{R}^d$, revisited

We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}ρ\nabla\cdot ρ\mathbb D\nabla +V$ on domains $Ω\subset \mathbb{R}^d$ with $\mathcal{C}^2$-boundary $\partial Ω$ and for large classes of coefficients $ρ,\; \mathbb{D}$ and $V$. We give criteria showing how the behavior as $x \rightarrow \partial Ω$ of these coefficients balances to ensure essential self-adjointness of $H$. On the way we prove a weak anisotropic Hardy inequality which is of independent interest.

非阿贝尔布伦-闵可夫斯基不等式 II

A nonabelian Brunn-Minkowski inequality II

该研究证明非紧李维数为n的单模局部紧群满足严格非阿贝尔布伦-闵可夫斯基不等式，建立了一般形式并证实相关猜想，还应用得到非紧型对称空间的等周不等式。

Minkowski二次不等式的极值

The Extremals of Minkowski's Quadratic Inequality

本文通过将任意凸体的混合体积表示为与高度退化椭圆算子相关联的Dirichlet形式，利用算子的定量刚性性质，完全解决了Minkowski二次不等式的极值问题并证实了R. Schneider的猜想。

A sharp centro-affine isospectral inequality of Szegö--Weinberger type and the $L^p$-Minkowski problem

We establish a sharp upper-bound for the first non-zero even eigenvalue (corresponding to an even eigenfunction) of the Hilbert-Brunn-Minkowski operator associated to a strongly convex $C^2$-smooth origin-symmetric convex body $K$ in $\mathbb{R}^n$. Our isospectral inequality is centro-affine invariant, attaining equality if and only if $K$ is a (centered) ellipsoid; this is reminiscent of the (non affine invariant) classical Szegö--Weinberger isospectral inequality for the Neumann Laplacian. The new upper-bound complements the conjectural lower-bound, which has been shown to be equivalent to …

容量型 John-Nirenberg 不等式再探

The Capacitary John-Nirenberg Inequality Revisited

本文研究平移不变 Hausdorff 容度的容量型 John-Nirenberg 不等式，揭示填充条件是弥补容量积分非线性的关键，并证明其等价于外容量与诱导 Hausdorff 容度等价，从而推广相关理论。

关于内在体积的 $L_p$-Brunn-Minkowski 不等式及 $L_p$-Christoffel-Minkowski 问题的注记

A note on the $L_p$-Brunn-Minkowski inequality for intrinsic volumes and the $L_p$-Christoffel-Minkowski problem

本文改进了 $L_p$-Brunn-Minkowski 不等式并证明了 $L_p$-Christoffel-Minkowski 问题的唯一性结果。

Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_ϕ$ affine and geominimal surface areas

This paper aims to develop basic theory for the dual Orlicz $L_ϕ$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequality, cyclic inequality, Santaló style inequality and Alexander-Fenchel type inequality are established.

Minkowski inequality for nearly spherical domains

We investigate the validity and the stability of various Minkowski-like inequalities for $C^1$-perturbations of the ball. Let $K\subseteq\mathbb R^n$ be a domain (possibly not convex and not mean-convex) which is $C^1$-close to a ball. We prove the sharp geometric inequality $$ \left(\int_{\partial K} \lVert II\rVert_1 d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_1(n)Per(K)^{\frac1{n-1}} , $$ where $C_1(n)$ is the constant that yields the equality when $K=B_1$ (and $\lVert II\rVert_1$ is the sum of the absolute values of the eigenvalues of the second fundamental form $II$ of $\partial K$). Mor…

Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures

We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman's reverse Brunn-Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman's deep technology of $M$-ellip…

Functional Brunn-Minkowski Inequalities Induced by Polarity

We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geom…

Poincaré不等式与分离集Minkowski内容的一种几何方法

A geometric approach to Poincaré inequality and Minkowski content of separating sets

本文用位置函数与分离比等几何工具，给出Poincaré不等式和分离集Minkowski内容下界关系的新短证，并推广至测度图。

非阿贝尔Brunn-Minkowski不等式

A nonabelian Brunn-Minkowski inequality

该研究将Brunn-Minkowski不等式推广到非阿贝尔局部紧群，证明其对无螺旋群（含实线性代数群、半单Lie群等）成立且锐利，采用维数归纳、最大紧子群分析和按比例平均技巧。

An inequality for the compositions of convex functions with convolutions and an alternative proof of the Brunn-Minkowski-Kemperman inequality

Let $m(G)$ be the infimum of the volumes of all open subgroups of a unimodular locally compact group $G$. Suppose integrable functions $ϕ_1 , ϕ_2 \colon G \to [0,1]$ satisfy $\| ϕ_1 \| \leq \| ϕ_2 \|$ and $\| ϕ_1 \| + \| ϕ_2 \| \leq m (G)$, where $\| \cdot \|$ denotes the $L^1$-norm with respect to a Haar measure $dg$ on $G$. We have the following inequality for any convex function $f \colon [0, \| ϕ_1 \| ] \to \mathbb{R}$ with $f(0) = 0$: \begin{align*} \int_{G}^{} f \circ ( ϕ_1 * ϕ_2 ) (g) dg \leq 2 \int_{0}^{\| ϕ_1 \|} f(y) dy + ( \| ϕ_2 \| - \| ϕ_1 \| ) f( \| ϕ_1 \| ). \end{align*} As a co…

On the equality conditions of the Brunn-Minkowski theorem

This article describes a new proof of the equality condition for the Brunn-Minkowski inequality.

Poincaré and Brunn--Minkowski inequalities on the boundary of weighted Riemannian manifolds

We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski inequality, thereby precluding any extensions beyond the Euclidean setting. A dual version for generalized mean-convex boundaries is also obt…

On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with vertex at the origin $o$ and having nonempty interior. The set $A\subset C$ is $C$-coconvex if the volume of $A$ is finite and $A^{\bullet}=C\setminus A$ is a closed convex set. For $0<p<1$, the $p$-co-sum of $C$-coconvex sets is introduced, and the corresponding $L_p$ Brunn-Minkowski inequality for $C$-coconvex sets is established. We also define the $L_p$ surface area measures, for $0\neq p\in \mathbb{R}$, of certain $C$-coconvex sets, which are critical in deriving a variational formula of the volume of the Wulff shape associate…

Fractional generalizations of Young and Brunn-Minkowski inequalities

A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It is shown that the generalized Brunn-Minkowski conjecture is true for convex sets; an application of this to the law of large numbers for random sets is described.