منابع مشابه
Volumes of Convex Lattice Polytopes
We show by a direct construction that there are at least exp{cV (d−1)/(d+1)} convex lattice polytopes in R of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family Pd(r) (to be defined in the text) of convex lattice polytopes whose volumes are between 0 and r/d!. Namely ...
متن کاملVolumes of Symmetric Random Polytopes
UK,N = 1 |K| | conv{x1, . . . , xN}|, where |A| denotes the volume of a Borel set A ⊂ R, and x1, . . . , xN are independent random points uniformly distributed in K. That is, UK,N is the normalized volume of a random polytope in K; in particular, UK = UK,n+1 is the normalized volume of a random simplex in K. Note that the distribution of UK,N is an affine invariant of K. For K ∈ Ks and N ≥ n, w...
متن کاملMatroid Polytopes and their Volumes
We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of M . Our proofs are based on a natural...
متن کاملExpectation of intrinsic volumes of random polytopes
Let K be a convex body in Rd, let j ∈ {1, . . . , d − 1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C2 +, then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C3 +) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to ...
متن کاملDependent Randomized Rounding for Matroid Polytopes and Applications
Motivated by several applications, we consider the problem of randomly rounding a fractional solutionin a matroid (base) polytope to an integral one. We consider the pipage rounding technique [5, 6, 36] andalso present a new technique, randomized swap rounding. Our main technical results are concentrationbounds for functions of random variables arising from these rounding techniques...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2004
ISSN: 0024-3795
DOI: 10.1016/j.laa.2003.09.005