On moments of a polytope (original) (raw)

Moment Varieties of Measures on Polytopes

ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA - CLASSE DI SCIENZE

The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated than in these two special cases. They offer challenges for both numerical and symbolic computing in algebraic geometry.

The inverse moment problem for convex polytopes

2012

We present a general and novel approach for the reconstruction of any convex d-dimensional polytope P , assuming knowledge of finitely many of its integral moments. In particular, we show that the vertices of an N-vertex convex polytope in R d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure of degree D), in d + 1 distinct directions in general position. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry of polytopes, combined with what is variously known as Prony's method, or the Vandermonde factorization of finite rank Hankel matrices.

Measures of Dirichlet type on regular polygons and their moments

Journal of Approximation Theory, 1992

The (k-I)-dimensional simplex is projected onto the convex hull of the kth roots of unity in C, and a dihedral-group-invariant Dirichlet-type measure is thereby constructed. The integrals of monomials z'"?" are obtained as single sums. A certain radial measure on the disc is obtained as a weak-*limit.

On the convex hull of uniform random points in a simpled-polytope

Discrete & Computational Geometry, 1991

Denote the expected number of facets and vertices and the expected volume of the convex hull Pn of n random points, selected independently and uniformly from the interior of a simple d-polytope by En(f), E.(v), and E~(V), respectively. In this note we determine the sharp constants of the asymptotic expansion of En(f), E.(v), and En(V), as n tends to infinity. Further, some results concerning the expected shape of P~ are given.

On the Graph-Density of Random 0/1-Polytopes

Lecture Notes in Computer Science, 2003

Let X d,n be an n-element subset of {0, 1} d chosen uniformly at random, and denote by P d,n := conv X d,n its convex hull. Let ∆ d,n be the density of the graph of P d,n (i.e., the number of one-dimensional faces of P d,n divided by n 2). Our main result is that, for any function n(d), the expected value of ∆ d,n(d) converges (with d → ∞) to one if, for some arbitrary ε > 0, n(d) ≤ (√ 2 − ε) d holds for all large d, while it converges to zero if n(d) ≥ (√ 2 + ε) d holds for all large d.

Uniform decompositions of polytopes

Applicationes Mathematicae, 2006

We design a method of decomposing convex polytopes into simpler polytopes. This decomposition yields a way of calculating exactly the volume of the polytope, or, more generally, multiple integrals over the polytope, which is equivalent to the way suggested in [9], based on Fourier-Motzkin elimination ([10, pp. 155-157]). Our method is applicable for finding uniform decompositions of certain natural families of polytopes. Moreover, this allows us to find algorithmically an analytic expression for the distribution function of a random variable of the form d i=1 c i X i , where (X 1 ,. .. , X d) is a random vector, uniformly distributed in a polytope.

On the Convex Hull of Random Points in a Polytope

Journal of Applied Probability, 1988

The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(log d–1 n) for any polytope, the expected number of vertices is Ω(log d–1 n) for any simple polytope, and the expected number of facets is O(log d–1 n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.

On polygonal measures with vanishing harmonic moments

Journal d'Analyse Mathématique, 2014

A polygonal measure is the sum of finitely many real constant density measures supported on triangles in C. Given a finite set S ⊂ C, we study the existence of polygonal measures spanned by triangles with vertices in S, which have all harmonic moments vanishing. For S generic, we show that the dimension of the linear space of such measures is |S|−3 2 .

On the isotropic constant of random polytopes

Advances in Geometry, 2000

Let K be an isotropic 1-unconditional convex body in R n . For every N > n consider N independent random points x1, . . . , xN uniformly distributed in K. We prove that, with probability greater than 1 − C 1 exp(−cn) if N ≥ c 1 n and greater than 1−C 1 exp(−cn/ log n) if n < N < c 1 n, the random polytopes KN := conv ± x1, . . . , ±xN and SN := conv{x1, . . . , xN } have isotropic constant bounded by an absolute constant C > 0.