arXiv subjects

David Yost

Publications and source records attributed to David Yost.

15 recordsLinked to original sources

arXiv2025

A refined lower bound theorem for $d$-polytopes with at most $2d$ vertices

In 1967, Gr\"unbaum conjectured that the function $$ \phi_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a $d$-polytope) with $d+s$ vertices. In 2021, Xue proved this conjecture for each $k\in[1\ldots d-2]$ and characterised the unique minimisers, each having $d+2$ facets. In this paper, we refine Xue's theorem by considering $d$-polytopes with $d+s$ vertices ($2\le s\le d$) and at least $d+3$ facets. If $s=2$, then there is precisely one minimiser for many values of $k$. For other values of $s$, the number of $k$-faces is at least $\phi_k(d+s,d)+\binom{d-1}{k}-\binom{d+1-s}{k}$, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$ and $k$. We also characterise the minimising polytopes.

math.CO↗

arXiv2024

A lower bound theorem for $d$-polytopes with $2d+2$ vertices

We establish a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, extending the previously known case for $k=1$. We identify all minimisers for $d\le 5$. Two distinct lower bounds emerge, depending on the number of facets of $P$. When $P$ has precisely $d+2$ facets, the lower bound is tight when $d$ is odd. If $P$ has at least $d+3$ facets, the lower bound is always tight, and equality holds for some $1\le k\le d-2$ only when $P$ has precisely $d+3$ facets. Moreover, for $1\le k\le \ceil{d/3}-2$, the minimisers among $d$-polytopes with $2d+2$ vertices have precisely $d+3$ facets, while for $\floor{0.4d}\le k\le d-1$, the lower bound arises from $d$-polytopes with $d+2$ facets.

math.CO↗

arXiv2024

Polytopes with low excess degree

We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as $2f_1-df_0$. We show that the excess degree of a $d$-polytope cannot lie in the range $[d+3,2d-7]$, complementing the known result that values in the range $[1,d-3]$ are impossible. In particular, many pairs $(f_0,f_1)$ are not realised by any polytope. For $d$-polytopes with excess degree $d-2$, strong structural results are known; we establish comparable results for excess degrees $d$, $d+2$, and $2d-6$. Frequently, in polytopes with low excess degree, say at most $2d-6$, the nonsimple vertices all have the same degree and they form either a face or a missing face. We show that excess degree $d+1$ is possible only for $d=3,5$, or $7$, complementing the known result that an excess degree $d-1$ is possible only for $d=3$ or $5$.

math.CO↗

arXiv2021

A lower bound theorem for $d$-polytopes with $2d+1$ vertices

The problem of calculating exact lower bounds for the number of $k$-faces of $d$-polytopes with $n$ vertices, for each value of $k$, and characterising the minimisers, has recently been solved for $n\le2d$. We establish the corresponding result for $n=2d+1$; the nature of the lower bounds and the minimising polytopes are quite different in this case. As a byproduct, we also characterise all $d$-polytopes with $d+3$ vertices, and only one or two edges more than the minimum.

math.CO↗

arXiv2021

Lower bound results for conditionally decomposable polytopes

It is possible for a combinatorial type of polytope to have both decomposable and indecomposable realizations; here decomposability is meant with respect to Minkowski addition. Such polytopes are called conditionally decomposable. We show that the minimum number of vertices of a conditionally decomposable $d$-polytope is in the range $[3d-3, 4d-4]$, and that for a polytope having a line segment for a summand, $4d-4$ is sharp. As an application, the exact lower bound of the number of $k$-faces of a decomposable $d$-polytope with $2d+m$ vertices ($2 \le m\le d-4$) is obtained. Concerning the facets, in dimension 4, the minimum number of facets of a conditionally decomposable polytope is 9, and in dimension $d\ge 5$, the minimum is $d+4$.

math.CO↗

arXiv2020

Minimum number of edges of polytopes with 2d + 2 vertices

We define an analogue of the cube and an analogue of the 5-wedge in higher dimensions, each with $2d+2$ vertices and $d^2+2d-3$ edges. We show that these two are the only minimisers of the number of edges, amongst d-polytopes with $2d+2$ vertices, for all $d$ except 4, 5 and 7. We also show that there are four sporadic minimisers in these low dimensions. We announce a partial solution to the corresponding problem for polytopes with $2d + 3$ vertices.

math.CO↗

arXiv2019

Lower bound theorems for general polytopes

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Grünbaum, for these values of $m$. For $v=2d+1$, we solve the same problem when $m=1$ or $d-2$; the solution was already known for $m= d-1$. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of $m$-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.

math.CO↗

arXiv2018

Polytopes close to being simple

It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$-polytopes with at most $d-2$ nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2 and $d-2$, showing that certain polytopes with more than two nonsimple vertices are reconstructible from their graphs. In particular, we prove that reconstructibility from graphs also holds for $d$-polytopes with $d+k$ vertices and at most $d-k+3$ nonsimple vertices, provided $k\ge 5$. For $k\le4$, the same conclusion holds under a slightly stronger assumption. Another measure of deviation from simplicity is the {\it excess degree} of a polytope, defined as $ξ(P):=2f_1-df_0$, where $f_k$ denotes the number of $k$-dimensional faces of the polytope. Simple polytopes are those with excess zero. We prove that polytopes with excess at most $d-1$ are reconstructible from their graphs, and this is best possible. An interesting intermediate result is that $d$-polytopes with less than $2d$ vertices, and at most $d-1$ nonsimple vertices, are necessarily pyramids.

math.CO↗

arXiv2018

Almost simplicial polytopes I. The lower and upper bound theorems

We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$ and $s$, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case of $s=0$. We characterize the minimizers and provide examples of maximizers, for any $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.

math.CO↗

arXiv2018

On the reconstruction of polytopes

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its $1$-skeleton. Call a vertex of a $d$-polytope \emph{nonsimple} if the number of edges incident to it is more than $d$. We show that (1) the face lattice of any $d$-polytope with at most two nonsimple vertices is determined by its $1$-skeleton; (2) the face lattice of any $d$-polytope with at most $d-2$ nonsimple vertices is determined by its $2$-skeleton; and (3) for any $d>3$ there are two $d$-polytopes with $d-1$ nonsimple vertices, isomorphic $(d-3)$-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for $4$-polytopes.

math.CO↗

arXiv2018

The excess degree of a polytope

We define the excess degree $ξ(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote the number of vertices and edges, respectively. This parameter measures how much $P$ deviates from being simple. It turns out that the excess degree of a $d$-polytope does not take every natural number: the smallest possible values are $0$ and $d-2$, and the value $d-1$ only occurs when $d=3$ or 5. On the other hand, for fixed $d$, the number of values not taken by the excess degree is finite if $d$ is odd, and the number of even values not taken by the excess degree is finite if $d$ is even. The excess degree is then applied in three different settings. It is used to show that polytopes with small excess (i.e. $ξ(P)<d$) have a very particular structure: provided $d\ne5$, either there is a unique nonsimple vertex, or every nonsimple vertex has degree $d+1$. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Secondly, we characterise completely the decomposable $d$-polytopes with $2d+1$ vertices (up to combinatorial equivalence). And thirdly all pairs $(f_0,f_1)$, for which there exists a 5-polytope with $f_0$ vertices and $f_1$ edges, are determined.

math.CO↗

arXiv2017

Chebyshev multivariate polynomial approximation and point reduction procedure

The theory of Chebyshev (uniform) approximation for univariate polynomial and piecewise polynomial functions has been studied for decades. The optimality conditions are based on the notion of alternating sequence. However, the extension the notion of alternating sequence to the case of multivariate functions is not trivial. The contribution of this paper is two-fold. First of all, we give a geometrical interpretation of the necessary and sufficient optimality condition for multivariate approximation. These optimality conditions are not limited to the case polynomial approximation, where the basis functions are monomials. Second, we develop an algorithm for fast necessary optimality conditions verifications (polynomial case only). Although, this procedure only verifies the necessity, it is much faster than the necessary and sufficient conditions verification. This procedure is based on a point reduction procedure and resembles the univariate alternating sequence based optimality conditions. In the case of univariate approximation, however, these conditions are both necessary and sufficient. Third, we propose a procedure for necessary and sufficient optimality conditions verification that is based on a generalisation of the notion of alternating sequence to the case of multivariate polynomials.

math.NA↗

arXiv2017

Compact convex sets with prescribed facial dimensions

While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the questions arising from this approach are highly nontrivial and give examples of convex sets for which the sets of extreme points have fractal dimension.

math.MG↗

arXiv2016

More indecomposable polyhedra

We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension $d\neq 2$, we show that of all the polytopes with $d^2+\frac{d}{2}$ or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.

math.CO↗

arXiv2015

Chebyshev approximation for multivariate functions

In this paper, we derive optimality conditions (Chebyshev approximation) for multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions is very elegant. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not very straightforward, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.

math.OC↗