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Register a new userLower bounds for the simplexity of the n-cube
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Authors
Alexey Glazyrin
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In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of $n$-dimensional cubes. In particular we show that the number of simplices in dissections of $n$-cubes without additional vertices is at least $(n+1)^{frac {n-1} 2}$.
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We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension $n$ and cardinality $N$ not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal.
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds are optimal for absolutely monotone potentials in the sense that for the linear programming technique they cannot be improved by using polynomials of the same or lower degree. When additional information about the structure (upper and lower bounds for the inner products) of the designs is known, improvements on the bounds are obtained. Furthermore, we provide `test functions for determining when the linear programming lower bounds for energy can be improved utilizing higher degree polynomials. We also provide some asymptotic results for these energy bounds.
We give a simple quantitative condition, involving the mapping content of Azzam--Schul, that implies that a Lipschitz map from a Euclidean space to a metric space must be close to factoring through a tree. Using results of Azzam--Schul and the present authors, this gives simple checkable conditions for a Lipschitz map to have a large piece of its domain on which it behaves like an orthogonal projection. The proof involves new lower bounds and continuity statements for mapping content, and relies on a qualitative version of the main theorem recently proven by Esmayli--Haj{l}asz.
Let $D$ denote the distance matrix for an $n+1$ point metric space $(X,d)$. In the case that $X$ is an unweighted metric tree, the sum of the entries in $D^{-1}$ is always equal to $2/n$. Such trees can be considered as affinely independent subsets of the Hamming cube $H_n$, and it was conjectured that the value $2/n$ was minimal among all such subsets. In this paper we confirm this conjecture and give a geometric interpretation of our result which applies to any subset of $H_n$.
The average kissing number of $mathbb{R}^n$ is the supremum of the average degrees of contact graphs of packings of finitely many balls (of any radii) in $mathbb{R}^n$. We provide an upper bound for the average kissing number based on semidefinite programming that improves previous bounds in dimensions $3, ldots, 9$. A very simple upper bound for the average kissing number is twice the kissing number; in dimensions $6, ldots, 9$ our new bound is the first to improve on this simple upper bound.
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