Related papers: Discrete intrinsic volumes

We provide a sharp double-sided estimate for Poincar\'e-Sobolev constants on a convex set, in terms of its inradius and $N-$dimensional measure. Our results extend and unify previous works by Hersch and Protter (for the first eigenvalue)…

Let Delta be an integral convex polytope containing the origin of dimension n in the n-dim real space and it is simplicial at all origin-less facets. Let A(Delta) be the space of all Laurent polynomials f parametered by its coefficients…

The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is…

We prove the surjectivity part of Goncharov's depth conjecture. We also show that the depth conjecture implies that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function…

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $\xi_1,...,\xi_d$ with covariance matrices $\Sigma_1,...,\Sigma_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of…

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that…

This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of…

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the…

For d-dimensional irrational ellipsoids E with d >= 9 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order o(r^{d-2}). The estimate refines an earlier authors'…

Intrinsic $L_p$ metrics are defined and shown to satisfy a dimension--free bound with respect to the Hausdorff metric.

We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex…

Let $G$ be a higher rank simple real algebraic group, or more generally, any semisimple real algebraic group with no rank one factors and $X$ the associated Riemannian symmetric space. For any Zariski dense discrete subgroup $\Gamma<G$, we…

Consider a polynomial $f$ with a convenient Newton polytope $P$ and generic complex coefficients. By the global version of the Kouchnirenko formula, the hypersurface $\{f = 0\} \subset \mathbb{C}^n$ has the homotopy type of a bouquet of…

Let $\Delta_n$ and $Q_n$ denote the regular $n$-simplex of side length $\sqrt{2}$ embedded in $\mathbb{R}^{n+1}$ and the volume one cube in $\mathbb{R}^n$, respectively. We derive a closed-form formula for the hyperplane volume projections…

The Minkowski mixed volume of $n$ subpolytopes $D_1, \dots, D_n$ of a polytope $P \subset {\mathbb R}^n$ clearly does not exceed the normalized volume $n! \text{Vol}(P)$. Equality holds if and only if the subpolytopes are interlaced, i.e.,…

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic…

Let $\mathcal{P}$ be an $n$-point subset of Euclidean space and $d\geq 3$ be an integer. In this paper we study the following question: What is the smallest (normalized) relative change of the volume of subsets of $\mathcal{P}$ when it is…

We describe the computation of polytope volumes by descent in the face lattice, its implementation in Normaliz, and the connection to reverse-lexicographic triangulations. The efficiency of the algorithm is demonstrated by several high…

In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a full…