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Register a new userOn Kalais conjectures concerning centrally symmetric polytopes
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In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture. It is well-known that the three conjectures hold in dimensions d leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d geq 5.
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A two-step model for generating random polytopes is considered. For parameters $d$, $m$, and $p$, the first step is to generate a simple polytope $P$ whose facets are given by $m$ uniform random hyperplanes tangent to the unit sphere in $mathbb{R}^d$, and the second step is to sample each vertex of $P$ independently with probability $p$ and let $Q$ be the convex hull of the sampled vertices. We establish results on how well $Q$ approximates the unit sphere in terms of $m$ and $p$ as well as asymptotics on the combinatorial complexity of $Q$ for certain regimes of $p$.
We introduce the fatness parameter of a 4-dimensional polytope P, defined as phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes bounded? We describe and analyze a hyperbolic geometry construction that produces 4-polytopes with fatness phi(P)>5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite covering spaces of surfaces, we show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3-sphere.
The Stern poset $mathcal{S}$ is a graded infinite poset naturally associated to Sterns triangle, which was defined by Stanley analogously to Pascals triangle. Let $P_n$ denote the interval of $mathcal{S}$ from the unique element of row $0$ of Sterns triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $ngeq 1$ let begin{align*} L_n(q)&=2cdotleft(sum_{k=1}^{2^n-1}A_{P_k}(q)right)+A_{P_{2^n}}(q), end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $ngeq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(q)$ is divisible by $L_{2n}(q)$. In this paper we obtain a simple recurrence relation satisfied by $L_n(q)$ and affirmatively solve Stanleys conjectures. We also establish the asymptotic normality of the coefficients of $L_n(q)$.
We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture${}^dagger$, Lieb permanent dominance conjecture, Bapat and Sunder conjecture${}^dagger$ on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open. We also raise a few new questions for future study. (${}^dagger$conjectures have been recently settled negatively.)
For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures -- one regarding the maximum possible value of $cp(G) + cp(overline{G})$ (due to de Caen, ErdH{o}s, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $max_G cp(G) + cp(overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.
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