Geometric valuation theory
Monika Ludwig
该文综述几何赋值理论，聚焦凸体与函数空间中赋值的分类结果。
本文简要介绍几何赋值理论，重点讨论凸体上及函数空间上赋值的分类结果。
A brief introduction to geometric valuation theory is given. The focus is on classification results for valuations on convex bodies and on function spaces.
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It is shown that every continuous valuation defined on the $n$-dimensional star bodies has an integral representation in terms of the radial function. Our argument is based on the non-trivial fact that continuous valuations are uniformly continuous on bounded sets. We also characterize the continuous valuations on the $n$-dimensional star bodies that arise as the restriction of a measure on $\mathbb R^n$.
We show that a radial continuous valuation defined on the $n$-dimensional star bodies extends uniquely to a continuous valuation on the $n$-dimensional bounded star sets. Moreover, we provide an integral representation of every such valuation, in terms of the radial function, which is valid on the dense subset of the simple Borel star sets. We also show that every radial continuous valuation defined on the $n$-dimensional star bodies can be decomposed as a sum $V=V^+-V^-$, where both $V^+$ and $V^-$ are positive radial continuous valuations.
Valuations on Convex Bodies and Functions
本文介绍几何赋值理论，聚焦凸体与凸函数上的SL(n)不变及刚体运动不变赋值的分类结果。
Classifications of $\rm{SL}(n)$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified characterizations of the Laplace transform on convex bodies, $L_p$ moment bodies, $L_p$ difference bodies, and polar $L_p$ moment bodies ($L_{-p}$ intersection bodies). Using the new classifications, we also establish some Euler-type relations.
From valuations on convex bodies to convex functions
本文建立凸体赋值与凸函数赋值之间的几何框架，利用 McMullen 经典结果重新证明已知的 n-外延齐次赋值刻画，并给出 1-外延齐次赋值的新刻画。
We show that every radial continuous valuation $V:\mathcal S_0^n\rightarrow \mathbb R$ defined on the $n$-dimensional star bodies $\mathcal S_0^n$, and verifying $V(\{0\})=0$, can be decomposed as a sum $V=V^+-V^-$, where both $V^+$ and $V^-$ are positive radial continuous valuations on $\mathcal S_0^n$ with $V^+(\{0\})=V^-(\{0\})=0$. As an application, we show that radial continuous rotationally invariant valuations $V$ on $\mathcal S_0^n$ can be characterized as the applications on star bodies which can be written as $$V(K)=\int_{S^{n-1}}θ(ρ_K)dm,$$ where $θ:[0,\infty)\rightarrow \mathbb R$ …
Measure-valued valuations on star bodies
本文对R^n中星体上的弱*连续测度值估值进行了完全分类，得到了旋转等变测度值估值的积分表示和对偶面积测度的刻画。
Very recently J. Kotrbaty has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge- Riemann bilinear relations in the Kahler geometry. The goal of this note is to apply Kotrbaty's theorem to obtain a few apparently new inequalities for mixed volumes of convex bodies.
Non-continuous valuations on convex bodies and a new characterization of volume
本文通过自动连续性技术，给出了凸体估值的新特征化，并确定了n维凸体上有界平移不变估值的可能齐次度。
We characterize the positive radial continuous and rotation invariant valuations $V$ defined on the star bodies of $\mathbb R^n$ as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is, $$V(K)=\int_{S^{n-1}}θ(ρ_K)dm,$$ where $θ$ is a positive continuous function, $ρ_K$ is the radial function associated to $K$ and $m$ is the Lebesgue measure on $S^{n-1}$. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.
Zonal valuations on convex bodies
该研究建立了凸体上带状连续平移不变赋值的完整分类，用面积测度主值积分表示，借助Firey球冠体积不等式新加权版证明收敛性，改进了Minkowski赋值的卷积表示，并给出有限凸函数空间上相关赋值分类的新证明。
For valuations on convex bodies in Euclidean spaces, there is by now a long series of characterization and classification theorems. The classical template is Hadwiger's theorem, saying that every rigid motion invariant, continuous, real-valued valuation on convex bodies in $\mathbb{R}^n$ is a linear combination of the intrinsic volumes. For tensor-valued valuations, under the assumptions of isometry covariance and continuity, there is a similar classification theorem, due to Alesker. Also for the local extensions of the intrinsic volumes, the support, curvature and area measures, there are ana…
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…
Smooth valuations on convex bodies and finite linear combinations of mixed volumes
本文基于 Alesker 对 McMullen 猜想的解，证明了凸体上连续平移不变齐次赋值可由有限个混合体积线性组合一致逼近，且部分参数可固定为不依赖赋值的椭球体。
The Alesker-Bernig-Schuster theorem asserts that each irreducible representation of the special orthogonal group appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity. Moreover, the theorem describes in terms of highest weights which irreducible representations appear with multiplicity one. In this paper, we present a refinement of this result, namely the explicit construction of a highest weight vector in each irreducible subrepresentation. We then describe how important natural operations on v…
In this paper, we endow the space of continuous translation invariant valuation on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to non-smooth valuations, which are in the completion of the spaces of valuations with respect to these norms. The novelty of our approach lies in the fact that our proof does not rely on the general theory of wave fronts, but on geometric inequalities deduced from optimal transport …
A complete classification is established of Minkowski valuations on lattice polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are multiples of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined discrete Steiner point.
We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we introduce flag support measures of convex bodies via a local Steiner formula and derive some of the properties of these measures.
In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen's polytope algebra, with the universal property that continuous valuations on convex bodies correspond to continuous homomorphisms on the group of convex bodies. To study this group, we first obtain a version of McMullen polynomiality for valuations that take values not in fields or vector spaces, but in abelian groups. Using this, we are able to equip the group of convex bodies with a grading that consists of real vector spaces in all positive degrees, mir…
A classification of SL$(n)$ contravariant, continuous function valued valuations on convex bodies is established. Such valuations are natural extensions of SL$(n)$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ proj…
The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant Minkowski tensors that were introduced by McMullen and characterized by Alesker. In analogy to the characterization theorems of Hadwiger and Alesker, we give here a complete classification of all locally defined tensor measures on convex bodies that share with the local Minkowski tensors the basic geometric properties of isometry covariance and weak continu…
We study dually epi-translation invariant valuations on cones of convex functions containing the space of finite-valued convex functions. The existence of a homogeneous decomposition is used to associate a distribution to every valuation of this type similar to the Goodey-Weil embedding for translation invariant valuations on convex bodies. The relation between the valuation and its associated distribution is used to establish a notion of support for valuations. As an application, we show that there are no $\mathrm{SL}(n)$ or translation invariant valuations except constant valuations in this …
The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the author's irreducibility theorem and few prope…
Metrics from valuations on convex bodies
本文提出两种基于估值的偏差，并通过构造内在伪度量来研究其几何特性，揭示估值与几何之间的关系。
A classification of all continuous GL(n) equivariant Minkowski valuations on convex bodies in $\mathbb{R}^n$ is established. Together with recent results of F.E. Schuster and the author, this article therefore completes the description of all continuous GL(n) intertwining Minkowski valuations.
All continuous SL(n)-covariant $L_p$-Minkowski valuations defined on convex bodies are completely classified. The $L_p$-moment body operators turn out to be the nontrivial prototypes of such maps.
We provide a description of the space of continuous and translation invariant Minkowski valuations $Φ:\mathcal{K}^n\to\mathcal{K}^n$ for which there is an upper and a lower bound for the volume of $Φ(K)$ in terms of the volume of the convex body $K$ itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over $(n-1)$-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give…
Floating bodies for ball-convex bodies
本文在球凸体类中引入浮动体，通过其体积右导数定义相对仿射表面积，并证明该量为刚体运动不变量及上半连续赋值。
A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions
本文建立了一个关于凸函数上向量值估值的功能类比Klain-Schneider定理，对连续、平移协变且简单的估值进行分类。在额外的旋转等变假设下，特征化了动量向量的分析对应物以及一个新的上皮平移不变估值。前者作为一族函数内在矩的最高次算子出现，这些矩通过平移与函数内在体积相关联。后者代表了一类Minkowski向量的最高次算子，这些向量在本文中引入，它们在凸体上没有经典对应物，因为它们由于Minkowski关系而消失。此外，还获得了关于极值次数的同质估值的进一步分类结果。
The Minkowski tensors are valuations on the space of convex bodies in ${\mathbb R}^n$ with values in a space of symmetric tensors, having additional covariance and continuity properties. They are extensions of the intrinsic volumes, and as these, they are the subject of classification theorems, and they admit localizations in the form of measure-valued valuations. For these local tensor valuations, restricted to convex polytopes, a classification theorem has been proved recently, under the assumption of isometry covariance, but without any continuity assumption. This characterization result is…
All upper semicontinuous and SL(n) invariant valuations on convex bodies containing the origin in their interiors are completely classified. Each such valuation is shown to be a linear combination of the Euler characteristic, the volume, the volume of the polar body, and the recently discovered Orlicz surface areas.
A geometric decomposition for unitarily invariant valuations on convex functions
本文对 $\mathbb{C}^n$ 上凸函数空间中连续、对偶外延平移不变且 $\mathrm{U}(n)$-不变的估值进行了完全分类，证明其空间可按特殊子空间上的消失性质分解为子空间直和，并给出基于两族 Monge-Ampère 型算子主值积分的唯一表示。
We 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.
Lefschetz operators on convex valuations
本文研究 Alesker 的 Lefschetz 算子作用于凸体上平移不变赋值的行为，对标量值赋值用 Radon 型变换刻画其作用，对旋转等变 Minkowski 赋值证明生成函数经卷积变换且卷积核满足 Legendre 型微分方程并严格正、除一点外光滑。
Polynomial valuations on convex functions and their maximal extensions
该研究探讨凸函数不同锥上的多项式赋值扩张问题，证明其可归结为赋值支集的几何阻碍，借助齐次分解与分布支集性质推导结论，并给出最高次赋值的显式积分表示。
Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
本文证明了凸体上光滑平移不变 valuations 的硬 Lefschetz 定理和 Hodge-Riemann 关系，并由此得到 McMullen 二次不等式对光滑严格正曲率边界凸体的成立。
The Vectorial Hadwiger Theorem on Convex Functions
本文完成了凸函数上满足对偶上边缘平移不变与旋转等变的赋值的分类，刻画了函数性 Minkowski 向量，并给出奇异密度及混合 Monge-Ampère 测度等表示，同时推广到超强制情形。
Flag measures are descriptors of convex bodies $K$ in $d$-dimensional Euclidean space generalizing the classical area measures. They have been used to provide general integral formulas for mixed volumes (see Hug, Rataj and Weil (2017)). Here, we consider an image measure $γ_j(K,\cdot)$ of flag measures, defined on the Grassmannian $G(d,j)$ of affine $j$-spaces, $1\le j\le d-1$, and show that it determines centrally symmetric bodies $K$ of dimension $\geq j+1$ uniquely. We then explain that Grassmann measures appear in the representation of smooth, translation invariant, continuous and even val…
Valuations in affine convex geometry
本文研究凸几何中满足保体积线性不变、连续且有限可加的估值构造，发现平移不变性下无新例，否则存在大量例子。
In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures.
Ball-convex bodies and $L_p$ relative surface areas
本文提出了一种新的球凸体表面积测度，通过加权球浮体的导数给出了 $L_p$ 相对表面积的几何解释，并引入了新的熵概念。
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of…
We give geometric interpretations of certain affine invariants of convex bodies. The affine invariants are the p-affine surface areas introduced by Lutwak. The geometric interpretations involve generalizations of the Santaló-bodies introduced by the authors in a previous paper.
We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of 2-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodie…
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the standard inner product, the Legendre transform associated with the given norm. This approach yields re-interpretations of various properties of convex functions, and new relations between such functions and geometric properties of the studied norm are also derived.
Translation invariant area measures on convex bodies
本文引入连续平移不变面积测度空间，证明GL(n,R)-光滑面积测度等于由法向环积分得到的面积测度，并给出紧致群等变分类结果及不变子模密度准则。
Mixed Christoffel-Minkowski problems for bodies of revolution
针对轴对称凸体的混合Christoffel-Minkowski问题，提出变换混合面积测度与体积的简便方法，改进Firey的面积测度局部估计，无需正则性条件给出完整解答，还得到一族轴对称凸估值的Hadwiger型定理。
The semigroup of convex bodies in ${\mathbb R}^n$ with Minkowski addition has a canonical embedding into an abelian group; its elements have been called virtual convex bodies. Geometric interpretations of such virtual convex bodies have been particularly fruitful under the restriction to polytopes. For general convex bodies, mainly the planar case has been studied, as a part of the more general investigation of hedgehogs. Here we restrict ourselves to strictly convex bodies in ${\mathbb R}^n$. A particularly natural geometric interpretation of virtual convex bodies can then be seen in the set …
The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann-Petty centroid inequality.
Superforms, supercurrents and convex geometry
本研究发展超形式演算作为凸几何工具，将其应用于凸体相关的赋值、不等式及方程，为凸几何研究提供新方法。
On a generalization of Godbersen's conjecture
本研究提出Godbersen猜想的推广形式，将Schneider对Rogers-Shephard不等式的推广拓展至高阶差体，证明其对反块凸体成立，并建立了与高阶混合体积及Alesker乘积相关的等价表述。
In this note we prove two results on the quantitative illumination parameter f(d) of the unit ball of a d-dimensional normed space introduced by K. Bezdek (1992). The first is that f(d) = O(2^d d^2 log d). The second involves Steiner minimal trees. Let v(d) be the maximum degree of a vertex, and s(d) of a Steiner point, in a Steiner minimal tree in a d-dimensional normed space, where both maxima are over all norms. F. Morgan (1992) conjectured that s(d) <= 2^d, and D. Cieslik (1990) conjectured v(d) <= 2(2^d-1). We prove that s(d) <= v(d) <= f(d) which, combined with the above estimate of f(d)…
We find general geometric conditions on a convex body of revolution K, in dimensions four and six, so that its intersection body IK is not a polar zonoid. We exhibit several examples of intersection bodies which are are not polar zonoids.
Centroids and equilibrium points of convex bodies
本文综述凸体形心与静平衡点相关几何问题的研究成果，涵盖两类形心不等式相关结果、凸体的平衡点分类、平衡点的位置结构与数量及凸多面体静平衡性质等内容。
Complementary bodies in sphere packing
本文研究立方密堆积中空隙形成的球截多面体互补体，推导其表面积与体积比，并证明截断四面体和八面体如何铺砌球间空隙，深化了对经典堆积问题及其几何互补体的理解。
We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result for convex bodies based on geometric moments. It turns out that the stability result is improved considerably by using another set of moments, namely Legendre moments. We present a reconstruction …
We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.
We study relations of some classes of $k$-convex, $k$-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{$k$-circular convex} and \textrm{$k$-circular visible} ones. Investigation of these bodies more general than $k$-convex and $k$-visible ones allows us to generalize some classical results of geometric tomography and find their new applications.
The Fourier transform of planar convex bodies and discrepancy over intervals of rotations
本文研究了平面凸体特征函数在仿射变换下平均的Fourier变换，建立了基于几何性质的上下界，并给出了仿射二次偏差渐近行为的精确结果，回答了Bilyk和Mastrianni的公开问题。
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