On 0-1 Polytopes with Many Facets

Abstract

There exist n-dimensional 0-1 polytopes with as many as ( cn log n ) nÂ4 facets. This is our main result. It answers a question of Komei Fukuda and Gu nter M. Ziegler.

Random Spaces Generated by Vertices of the Cube

“…In the range N ≥ exp((log n) 2 ), the first inclusion was recently proved by Bárány and Pór in [4] (see the remarks after Theorem 2.2). Fact 2 should be compared with the following result which was proved in [17]: There exists a constant κ > 0 with the Random Spaces Generated by Vertices of the Cube 257 following property: for every δ ∈ (0, 1) and every convex body K with centroid at the origin in R n , N ≥ c(δ)n κ points x 1 , .…”

“…In a very recent paper, Bárány and Pór [4] showed the existence of 0-1 polytopes with superexponential number of facets. One main step in their argument is a statement equivalent to Theorem 2.1 (Lemma 4.3 in [4]) which is proved by a refinement of the method of [12] for the range N ≥ exp((log n)…”

Stability Properties of Neighbourly Random Polytopes

“…[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].…”

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

“…We study lower bounds. A major breakthrough in this direction was made by Bárány and Pór in [1]; they proved that g(n) ≥ cn log n n /4 , ( 1.3) where c > 0 is an absolute constant. We show that the exponent n/4 can in fact be improved to n/2: It is interesting to compare this estimate with the known bounds for the expected number of facets of the convex hull P N ,n of N independent random points which are uniformly distributed on the sphere S n−1 .…”

“…Theorem 1.1 gives a lower bound which is "practically of this order": for every ε > 0 one has g(n) > n (0.5−ε)n (1.6) if n is large enough. The existence of 0/1 polytopes with many facets is established by a refinement of the probabilistic method developed in [1]. It is more convenient to work with ±1 polytopes (i.e., polytopes whose vertices are sequences of signs).…”