积分几何中的弦测度及其闵可夫斯基问题

On Explicit Holmes-Thompson Area Formula in Integral Geometry

In this article, we give an exposition on the Holmes-Thompson theory developed by Alvarez. The space of geodesics in Minkowski space has a symplectic structure which is induced by the projection from the sphere-bundle. we show that it can be also obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the tangent bundle of the Minkowski unit sphere. We give detailed descriptions and expositions on Holmes-Thompson volumes in Minkowski space by the symplectic structure and the Crofton measures for them. For the Minkowski plane, a normed two dimensional space, we …

三角形弦幂积分 I_2 的闭式

A Closed Form for the Chord-Power Integral I_2 of a Triangle

针对任意三角形，建立了弦幂积分 I_2 的闭式表达式，包含边长的对称对数，并推导出幂级数表示和等周不等式，其中等边三角形唯一达到最优常数。

整格的逆向 Minkowski 定理的一个简单证明

A simple proof of a reverse Minkowski theorem for integral lattices

本文用简单证明建立整格中给定平方范数向量数目的二项式上界，得到近乎紧的逆向 Minkowski 型定理。

Minkowski sum of polytopes and its normality

In this paper, we consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex polytopes. We also study Minkowski sums of edge polytopes and give a sufficient condition for Minkowski sums of edge polytopes to have IDP.

A characterization of some mixed volumes via the Brunn-Minkowski inequality

We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$ defined as follows: ${\mathcal F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body $K$. We prove that if $\mathcal F$ satisfies an inequality of Brunn--Minkowski type, then $f$ is the support function of a convex body, i.e., $\mathcal F$ is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree $n-1$ and satisfy a Brunn--Minkowski type inequality.

Lutwak-Petty projection inequalities for Minkowski valuations and their duals

Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for inte…

Orlicz version of the mixed width integrals

In the paper, our main aim is to generalize the width integrals to the Orlicz space. Under the framework of Orlicz Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating Orlicz first order variation of the width integrals, and call as Orlicz mixed width integrals. The fundamental notions and conclusions of the width integrals and Minkoswki and Brunn-Minkowski inequalities for the width integrals are extended to an Orlicz setting and the related concepts and inequalities of Lp mixed width integrals of convex body are also derived.

Even Minkowski Valuations

A new integral representation of smooth translation invariant and rotation equivariant even Minkowski valuations is established. Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived. Moreover, the action of Alesker's Hard Lefschetz operators on these Minkowski valuations is explored in detail.

对数凹函数的Riesz $α$-能量及相关Minkowski问题

The Riesz $α$-energy of log-concave functions and related Minkowski problem

本文计算了对数凹函数Riesz $α$-能量的一阶变分并导出相应测度，提出了Riesz $α$-能量Minkowski问题，证明了其在光滑条件下可简化为Monge-Ampère型方程，并在温和条件下求解了该问题。

平面各向异性 Voronoi 图的 Minkowski 张量

Minkowski Tensors of Planar Anisotropic Voronoi Diagrams

针对平面各向异性 Voronoi 图，推导了星凸单元的任意阶 Minkowski 张量半解析公式，利用接触函数和微分同胚变换实现几何解析处理，并支持数值近似。

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…

The dual Minkowski problem for symmetric convex bodies

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved usi…

秩至多32的整幺模格的高斯质量的锐利反Minkowski不等式

A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$

通过模函数参数化和范数1分裂、ADE根计数及影子正性，证明秩不超过32的整幺模格的高斯质量在稳定格中由整数格最大化，并得到秩40偶幺模格的严格不等式。

Integral-Geometric Formulas for Perimeter in S^2, H^2, and Hilbert Planes

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central angle from an origin and the distance to the corresponding support line. The Minkowski formula for perimeter in E^2 involves polar coordinates and the geodesic curvature of the boundary of K. We generalize this to S^2 and H^2. In E^2 the Cauchy and Minkowski formulas are locally equ…

通过球面设计实现整格的反向闵可夫斯基壳层界中的等式

Equality in a Reverse Minkowski Shell Bound for Integral Lattices via Spherical Designs

本文通过球面设计理论，分类了Regev-Stephens-Davidowitz关于整格壳层计数不等式的等式情形，证明了仅当格为Z^n或E_8时等式成立。