Intrinsic Volumes of the Brownian Motion Body

Concentration of the Intrinsic Volumes of a Convex Body

The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic volume sequence concentrates sharply around a specific index, called the central intrinsic volume. Furthermore, among all convex bodies whose central intrinsic volume is fixed, an appropriately scaled cube has the intrinsic volume sequence with maximum entropy.

Intrinsic volumes of ellipsoids

We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that \begin{align*} V_k({\mathcal E})=κ_k\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\,\rm{d}t \end{align*} for all $k=1,\ldots,d$, where $s_{k-1}$ is the $(k-1)$-th elementary symmetric polynomial and $κ_k$ is the volume of the $k$-dimensional unit ball. Some examples of th…

An intrinsic volume metric for the class of convex bodies in $\mathbb{R}^n$

A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width …

Conic intrinsic volumes of Weyl chambers

We give a new, direct proof of the formulas for the conic intrinsic volumes of the Weyl chambers of types $A_{n-1}, B_n$ and $D_n$. These formulas express the conic intrinsic volumes in terms of the Stirling numbers of the first kind and their $B$- and $D$-analogues. The proof involves an explicit determination of the internal and external angles of the faces of the Weyl chambers.

随机β-prime多胞体的内蕴体积的高阶矩

Higher moments of intrinsic volumes of random beta-prime polytopes

研究β-prime多胞体（由β-prime分布随机点生成的凸包）的体积和内蕴体积的矩收敛性，证明其依分布收敛到具有幂律强度函数的泊松点过程凸包，并利用球面随机多胞体的矩收敛性。

Uniqueness of a 3D ellipsoid with given intrinsic volumes

Let $\mathcal{E}$ be an ellipsoid in $\mathbb{R}^n$. A. Gusakova and D. Zaporozhets conjectured that $\mathcal{E}$ is uniquely (up to rigid motions) determined by its intrinsic volumes. We prove this conjecture for $n = 3$.

Planar Projections and Second Intrinsic Volume

Consider random shadows of a cube and of a regular tetrahedron. Area and perimeter of the former are positively dependent (with correlation 0.915...), whereas area and perimeter of the latter appear to be negatively dependent. This is only one result of many, all involving generalizations of mean width.

Intrinsic volumes of random polytopes with vertices on the boundary of a convex body

Let $K$ be a convex body in $\R^d$, let $j\in\{1, ..., d-1\}$, and let $\varrho$ be a positive and continuous probability density function with respect to the $(d-1)$-dimensional Hausdorff measure on the boundary $\partial K$ of $K$. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $\partial K$ according to the probability distribution determined by $\varrho$. For the case when $\partial K$ is a $C^2$ submanifold of $\R^d$ with everywhere positive Gauss curvature, M. Reitzner proved an asymptotic formula for the expectation of the difference of the $j$th int…

From Steiner Formulas for Cones to Concentration of Intrinsic Volumes

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can th…

Orlicz球的内蕴体积的指数轮廓

Exponential profiles for the intrinsic volumes of Orlicz balls

该研究针对一大类Orlicz函数，求得阶与维度成比例的内蕴体积的指数速率，证明了相关指数轮廓猜想，扩展了ℓ_p球的对应结果，依赖大偏差理论完成证明。

Discrete intrinsic volumes

For a convex lattice polytope $P\subset \mathbb R^d$ of dimension $d$ with vertices in $\mathbb Z^d$, denote by $L(P)$ its discrete volume which is defined as the number of integer points inside $P$. The classical result due to Ehrhart says that for a positive integer $n$, the function $L(nP)$ is a polynomial in $n$ of degree $d$ whose leading coefficient is the volume of $P$. In particular, $L(nP)$ approximates the volume of $nP$ for large $n$. In convex geometry, one of the central notion which generalizes the volume is the intrinsic volumes. The main goal of this paper is to introduce their…

On $p$-Brunn-Minkowski inequalities for intrinsic volumes with $0\leq p<1$

We prove the validity of the $p$-Brunn-Minkowski inequality for the intrinsic volume $V_k$, $k=2,\dots, n-1$, of convex bodies in $\mathbb{R}^n$, in a neighborhood of the unit ball, for $0\le p<1$. We also prove that this inequality does not hold true on the entire class of convex bodies of $\mathbb{R}^n$, when $p$ is sufficiently close to $0$.

Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

We 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 which 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 conve…

Stochastic completeness of graphs: bounded Laplacians, intrinsic metrics, volume growth and curvature

The goal of this article is to survey various results concerning stochastic completeness of graphs. In particular, we present a variety of formulations of stochastic completeness and discuss how a discrepancy between uniqueness class and volume growth criteria in the continuous and discrete settings was ultimately resolved via the use of intrinsic metrics. Along the way, we discuss some equivalent notions of boundedness in the sense of geometry and of analysis. We also discuss various curvature criteria for stochastic completeness and discuss how weakly spherically symmetric graphs establish t…

Fan Valuations and spherical intrinsic volumes

We generalize valuations on polyhedral cones to valuations on fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion-restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion-restriction proof of a result of Klivans-Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generaliz…

Whitney numbers of arrangements via measure concentration of intrinsic volumes

We verify the Rota-Heron-Welsh conjecture for matroids realizable as c-arrangements: the coefficients of the characteristic polynomial of the associated matroid are log-concave. This family of matroids strictly contains that of complex hyperplane arrangements. Our proof combines the study of intrinsic volumes of certain extensions of arrangements and the Levy--Milman measure concentration phenomenon on realization spaces of arrangements.

内度量与外度量比值及体积

Ratio of Intrinsic Metric to Extrinsic Metric and Volume

本文研究了内度量与外度量比值与面积的关系，并给出了特定曲面比值最大值的下界，同时通过例子说明全局下界不存在。

Sharp phase transitions in Euclidean integral geometry

The intrinsic volumes of a convex body are fundamental invariants that capture information about the average volume of the projection of the convex body onto a random subspace of fixed dimension. The intrinsic volumes also play a central role in integral geometry formulas that describe how moving convex bodies interact. Recent work has demonstrated that the sequence of intrinsic volumes concentrates sharply around its centroid, which is called the central intrinsic volume. The purpose of this paper is to derive finer concentration inequalities for the intrinsic volumes and related sequences. T…

Random ball-polyhedra and inequalities for intrinsic volumes

We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the …

持久内蕴体积

Persistent Intrinsic Volumes

该研究结合几何测度论与持久同调工具，开发了从豪斯多夫接近的集合估计ℝᵈ紧子集内蕴体积的新方法，获内蕴体积稳定性结果，收敛速率为线性。

On convergence of intrinsic volumes of Riemannian manifolds

In 1939 H. Weyl has introduced the so called intrinsic volumes $V_i(M^n), i=0,\dots,n$, (known also as Lipschitz-Killing curvatures) for any closed smooth Riemannian manifold $M^n$. Given a Riemmanian submersion of compact smooth Riemannian manifolds $M\to B$, $B$ is connected. For $\varepsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $\varepsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_\varepsilon$. The main result says that $\lim_{\varepsi…

计算子水平集的内蕴体积及其应用

Computing intrinsic volumes of sublevel sets and applications

为凸多项式子水平集的内蕴体积和对偶体积建立了统一的Laplace-Grassmannian表示，并推导出Löwner-John型存在唯一性、分块分解原理和无坐标格点偏差界等推论。

Reverse and dual Loomis-Whitney-type inequalities

Various results are proved giving lower bounds for the $m$th intrinsic volume $V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in terms of the $m$th intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when $m=1$ and $m=n-1$. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume $V_1(K)$, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke …

凸体的幅度与Holmes-Thompson内蕴体积

Magnitude and Holmes-Thompson intrinsic volumes of convex bodies

本文证明了超度量赋范空间中凸体幅度可由Holmes-Thompson内蕴体积控制的上界，并借此给出Mahler猜想zonoid情形与Sudakov最小值不等式的新证明。

On the magnitude and intrinsic volumes of a convex body in Euclidean space

Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is closely related to intrinsic volumes. Here we prove an upper bound for the magnitude of a convex body in Euclidean space in terms of its intrinsic volumes. The result is deduced from an analogous known result for magnitude in $\ell_1^N$, via approximate embeddings of Euclidean space into high-dimensional $\ell_1^N$ spaces. As a consequence, we deduce a sufficient condition for infinite-dimensional …

The Weyl principle on the Finsler frontier

Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the coefficients of the tube volume polynomial are intrinsic invariants of the metric. As a consequence, the intrinsic volumes behave naturally under isometric immersions. This phenomenon, subsequently observed in a number of different geometric settings, is commonly referred to as the Weyl principle. In general normed spaces, the Holmes-Thompson intrinsic volumes …

关于函数内在体积及其扩展的不等式与反例

Inequalities and counterexamples for functional intrinsic volumes and beyond

本文研究了函数内在体积的不等式失效问题，通过反例和变分函数量建立了Wulff型不等式，并推广了Alexandrov-Fenchel不等式，恢复了Polya-Szegő型不等式的一个特例。

Conic support measures

The conic support measures localize the conic intrinsic volumes of closed convex cones in the same way as the support measures of convex bodies localize the intrinsic volumes of convex bodies. In this note, we extend the `Master Steiner formula' of McCoy and Tropp, which involves conic intrinsic volumes, to conic support measures. Then we prove Hölder continuity of the conic support measures with respect to the angular Hausdorff metric on convex cones and a metric on conic support measures which metrizes the weak convergence.

带边界流形在体积收敛且距离下有界时的内蕴平坦稳定性

Intrinsic Flat Stability of Manifolds with Boundary where Volume Converges and Distance is Bounded Below

本文针对带边界流形，在度量下界、直径有界、体积收敛及边界度量 $L^{\ rac{m-1}{2}}$ 收敛条件下，证明了体积保持的内蕴平坦收敛，并用于正质量定理的稳定性证明。

Properties of the Intrinsic Flat Distance

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio-Kirchheim's Slicing Theorem combined with an adapted version Gromov's Filling Volume.

Intrinsic metrics on graphs: A survey

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we dete…