A. Colesanti, M. Ludwig, F. Mussnig
The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.
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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 …
All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.
A classification of SL$(n)$ contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new SL$(n)$ covariant Minkowski valuation on convex functions is defined and characterized.
Upper semicontinuous valuations on convex functions of one variable
该论文在紧定义域的凸Lipschitz函数空间上，对满足上半连续、平移及对偶上镜平移不变性的赋值进行了完整分类。
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。
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。
The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere
本文证明了球面Lipschitz函数空间上连续对偶平移不变赋值可唯一分解为0、1、2次齐次赋值之和，并否定了3次及以上非平凡赋值的可能性，同时刻画了相应空间的稠密子空间。
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。
Equivariant Valuations on Convex Functions
本研究对满足对偶上图平移不变性、关于保体积线性映射等变或反变的有限凸函数空间连续赋值进行分类，找到差体映射的赋值论泛函类似物，证明投影体映射推广不存在且该结论适用于原点邻域有限的凸函数赋值。
All non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^n$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\operatorname{SL}(n)$ and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume and polar volume are characterized.
A Paley-Wiener-Schwartz Theorem for smooth valuations on convex functions
本文证明了凸函数上光滑赋值的 Paley-Wiener-Schwartz 定理，通过 Fourier-Laplace 变换刻画连续对偶上平移不变赋值，并以此推导平移群表示的光滑向量积分表示，最终实现闭和仿射不变子空间的完整分类及混合 Monge-Ampère 算子赋值的密度结果。
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 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…
We construct valuations on the space of finite-valued convex functions using integration of differential forms over the differential cycle associated to a convex function. We describe the kernel of this procedure and show that the intersection of this space of smooth valuations with the space of all continuous dually epi-translation invariant valuations on convex functions is dense in the latter. As an application, we obtain a description of 1-homogeneous, continuous, dually epi-translation invariant valuations that are invariant with respect to a compact subgroup operating transitively on the…
The Hadwiger theorem on convex functions, I
本文对 ${\mathbb R}^n$ 上超强制凸函数空间中所有连续、上图平移与旋转不变的对合进行了完整分类，通过引入奇异 Hessian 对合，将其定义为经典内蕴体积的函数版本。
The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant with respect to the orthogonal (or special orthogonal) group. Some applications to integral geometry are given.
A new class of continuous valuations on the space of convex functions on $\mathbb{R}^n$ is introduced. On smooth convex functions, they are defined for $i=0,\dots,n$ by \begin{equation*} u\mapsto \int_{\mathbb{R}^n} ζ(u(x),x,\nabla u(x))\,[\operatorname{D}^2 u(x)]_i\,{\rm d} x \end{equation*} where $ζ\in C(\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n)$ and $[\operatorname{D}^2 u]_i$ is the $i$th elementary symmetric function of the eigenvalues of the Hessian matrix, $\operatorname{D}^2 u$, of $u$. Under suitable assumptions on $ζ$, these valuations are shown to be invariant under translation…
There is a well known construction of weakly continuous valuations on convex compact polytopes in R^n. In this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the Hausdorff metric to all convex compact subsets of R^n. It is shown that there is a necessary condition on the initial data for such an extension. In the case of R^3 more explicit results are obtained.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
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.
We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. Among these valuations we prove integral representation formulas for those which are, additionally, simple or homogeneous.
Geometric valuation theory
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。
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。
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。
New proofs of the Hadwiger theorem for smooth and for continuous valuations on convex functions are obtained, and the Klain-Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on functions with lower dimensional domains is proved, and its connection to the Abel transform is explained.
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。
Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on the dual space $V^*$. This isomorphism commutes with the natural action of the full linear group on both spaces of valuations, maps product on the source (introduced in arXiv:math/0301148) to the convolution on the target (introduced in arXiv:math/0607449), and satisfies a Plancherel type inversion formula. As an application, a version of the hard Lefschetz t…
The Legendre transform, the Laplace transform and valuations
本文研究Legendre变换、Laplace变换在不同函数类上的特征化，揭示其作为反变估值的唯一性和对偶变换的性质。
We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation formula for those valuations which are N-homogeneous. Moreover, we introduce the notion of Klain's functions for these type of valuations.
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.
Unitarily invariant valuations on convex functions
研究复数空间中有限值凸函数的单位元不变估值，证明其可由特定子空间限制唯一确定。
A new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space $\HH^n$ is presented. In particular new examples of $Sp(n)Sp(1)$-invariant translation invariant continuous valuations are constructed. This method is based on the theory of plurisubharmonic functions of quaternionic variables.
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。
Singular valuations and the Hadwiger theorem on convex functions
本文刻画了光滑、旋转且对偶平移不变的赋值，借此给出凸函数上 Hadwiger 定理的新证明，并利用微分圈积分和 Hessian 测度的主值积分给出了泛函内蕴体积的新表示。
Monotone invariant valuations on convex cones
本文分类了闭凸锥空间上的单调旋转不变估值，证明其为具有递增系数的锥体内蕴体积线性组合，并解决了McMullen猜想。
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 …
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…
A classification of SL$(n)$ invariant valuations on the space of convex polytopes in $R^n$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $R^n$ that contain the origin.
A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.
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。
Tensor valuations on Sobolev spaces
本文对Sobolev空间上的连续、SL(n)逆变且平移不变的张量赋值进行了完整分类，并证明齐次情形下这些赋值恰为Fisher信息张量，即Fisher信息矩阵的高阶推广。
Two classes of measure-valued valuations on convex functions related to Monge-Ampère operators are investigated and classified. It is shown that the space of all valuations with values in the space of complex Radon measures on $\mathbb{R}^n$ that are locally determined, continuous, dually epi-translation invariant as well as translation equivariant, is finite dimensional. Integral representations of these valuations and a description in terms of mixed Monge-Ampère operators are established, as well as a characterization of $\mathrm{SO}(n)$-equivariant valuations in terms of Hessian measures.
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.
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。
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。
A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwining Minkowski valuations.
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…
Convex valuations, from Whitney to Nash
本文研究赋值的Whitney问题，证明当$S$为$r$-平面格拉斯曼流形且$r\boldsymbol{\rm \u2265}j+2$时相容性条件充分，补充了唯一性定理，还推导了紧流形上赋值的Nash型定理及Crofton公式存在性。
All measurable and $\operatorname{SL}(n)$-covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified. The moment vector is shown to be essentially the only such valuation.
Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets
本文在非阿基米德类比空间中引入乘积与卷积两种乘法结构，提出混合硬 Lefschetz 定理和 Hodge-Riemann 关系的猜想，并证明乘积的度 1 与卷积的余度 1 的非混合 Hodge-Riemann 关系。
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…
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…
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…
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…
Functional analogs of the Euler characteristic and volume together with a new analog of the polar volume are characterized as non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuations on the space of finite, convex and coercive functions on ${\mathbb R}^n$.
SL(n)-covariant symmetric tensor valuations on polytopes
该研究对n维欧氏空间凸多面体上的SL(n)协变对称张量赋值完成无连续性假设的分类，确定唯一此类赋值为矩张量。
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.
We describe various structures of algebraic nature on the space of continuous valuations on convex sets, their properties (like versions of Poincaré duality and hard Lefschetz theorem), and their relations and applications to integral geometry.
The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivial operator with these properties.
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。
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