Lower Bound for the Maximal Number of Facets of a 0/1 Polytope

Let µ be an even compactly supported Borel probability measure on the real line. For every N > n consider N independent random vectors X 1 , . . . , X N in R n , with independent coordinates having distribution µ. We establish a sharp threshold for the volume of the random polytope K N := conv X 1 , . . . , X N , provided that the Legendre transform λ of the cumulant generating function of µ satisfies the conditionwhere α is the right endpoint of the support of µ. The method and the result generalize work of Dyer, Füredi and McDiarmid on 0/1 polytopes. We verify ( * ) for a large class of distributions.

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“…This is a consequence of the next lemma (which essentially appears in [7], [2] and [9]). Lemma 4.1: Let 0 < r < λ(α).…”

Section: Threshold For the Volumementioning

confidence: 85%

“…The method they introduced in [7] proved to be extremely useful and accurate; for example, it played a key role in the approach introduced by Bárány and Pór in [2] in order to establish that there exist ±1 polytopes with a superexponential number of facets, which was further developed in [9] and [10].…”

Section: Introductionmentioning

confidence: 98%

Threshold for the volume spanned by random points with independent coordinates

Gatzouras

Giannopoulos²

2008

Isr. J. Math.

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“…In particular, if the 2 d vertices of the entire cube are ordered lexicographically then the total number of facets of all intermediate polytopes produced by an incremental convex hull algorithm to compute the cube is bounded from above by 3d · 2 d , while for an arbitrary (even for a random) ordering there might be intermediate polytopes with super-exponentially many vertices (due to the results of Bárány and Pór [2] and Gatzouras et al [5]). …”

Section: Incremental Convex-hull Algorithmsmentioning

confidence: 97%

“…They showed that a random d-dimensional 0/1-polytope with roughly 2 d/log 2 d vertices in expectation has at least (roughly) 2 (1/4)d log 2 d facets. Recently, this bound was even improved to 2 (1/2)d log 2 d by Gatzouras et al [5]. The best known upper bound currently is O((d − 2)!)…”

Section: Introductionmentioning

confidence: 94%

Revlex-initial 0/1-polytopes

Gillmann

Kaibel

2006

Journal of Combinatorial Theory, Series A

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We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers g nfac (d, n) of facets and the minimum average degree g avdeg (d, n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy g nfac (d, n) 3d and g avdeg (d, n) d + 4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.

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“…[4,8,14], see also the survey [9]). Random polytopes (including +1/−1 polytopes) also play a very important role in combinatorics; we mention only a few recent results [1][2][3].…”

Section: Introductionmentioning

confidence: 99%