A Polynomial Number of Random Points does not Determine the Volume of a Convex Body

Mixed volume of infinite-dimensional convex compact sets

Let $K$ be a convex compact $GB$-subset of a separable Hilbert space $H$. Denote by $\mathrm{Spec}_k K$ the set $\{(ξ_1(h), \ldots, ξ_k(h))\colon h\in K\}\subset \mathbb{R}^k,$ where $ξ_1, \ldots, ξ_k$ are independent copies of the isonormal Gaussian process on $H$. Tsirelson showed that in this case the intrinsic volumes of $K$ satisfy the relation \begin{equation*} V_k(K)= \frac{(2π)^{k/2}}{k!κ_k} \mathbf{E}\,\mathrm{Vol}_k(\mathrm{Spec}_k K). \end{equation*} Here, $\mathbf{E} \ \mathrm{Vol}_k(\mathrm{Spec}_k K)$ is the mean volume of $\mathrm{Spec}_k K$ and $κ_k$ is the volume of the $k$-di…

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 …

Random volumes in d-dimensional polytopes

Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Füredi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentia…

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.

Volumetric Properties of the Convex Hull of an n-dimensional Brownian Motion

Let K be the convex hull of the path of a standard brownian motion B(t) in R^n, taken at time 0 < t < 1. We derive formulas for the expected volume and surface area of K. Moreover, we show that in order to approximate K by a discrete version of K, namely by the convex hull of a random walk attained by taking B(t_n) at discrete (random) times, the number of steps that one should take in order for the volume of the difference to be relatively small is of order n^3. Next, we show that the distribution of facets of K is in some sense scale invariant: for any given family of simplices (satisfying s…

The mean width of random polytopes circumscribed around a convex body

Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an asymptotic formula for the expectation of the difference of the mean widths of $K^{(n)}$ and K, and another asymptotic formula for the expectation of the number of facets of $K^{(n)}$. These results are achieved by establishing an asymptotic result on weighted volume approximation of $K$ and by "dualizing" it using polarity.

Best and random approximation of a convex body by a polytope

In this paper, we give an overview of some results concerning best and random approximation of convex bodies by polytopes. We explain how both are linked and see that random approximation is almost as good as best approximation.

凸体避开随机点与缺项膨胀的最大体积

Maximal volume of convex bodies avoiding random points and lacunary dilates

该研究确定了缺项序列下凸体避开随机点的最大体积渐近公式，并证明缺项性条件不可被稀疏性替代。

On the monotonicity of the expected volume of a random simplex

Let a random simplex in a d-dimensional convex body be the convex hull of d+1 random points from the body. We study the following question: As a function of the convex body, is the expected volume of a random simplex monotone non-decreasing under inclusion? We show that this holds if d is 1 or 2, and does not hold if d >= 4. We also prove similar results for higher moments of the volume of a random simplex, in particular for the second moment, which corresponds to the determinant of the covariance matrix of the convex body. These questions are motivated by the slicing conjecture.

Confirmation of Matheron's conjecture on the covariogram of a planar convex body

The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffrac…

Volume Approximation of Strongly ${\mathbb C}$-Convex Domains by Random Polyhedra

Polyhedral-type approximations of convex-like domains in $\mathbb{C}^d$ have been considered recently by the second author. In particular, the decay rate of the error in optimal volume approximation as a function of the number of facets has been obtained. In this article, we take these studies further by investigating polyhedra constructed using random points (Poisson or binomial process) on the boundary of a strongly $\mathbb{C}$-convex domain. We determine the rate of error in volume approximation of the domain by random polyhedra, and conjecture the precise value of the minimal limiting con…

刻画具有非凸能量的梯度模型的无限体积极限

Characterising the infinite volume scaling limit of gradient models with non-convex energy

本文研究非凸能量梯度模型的无限体积极限，在低温有界倾斜条件下证明极限为表面张力黑塞矩阵为协方差的连续高斯自由场，补全了相关非凸能量猜想的证明。