scholarly journals Balanced generalized lower bound inequality for simplicial polytopes

2017 ◽  
Vol 24 (2) ◽  
pp. 1677-1689 ◽  
Author(s):  
Martina Juhnke-Kubitzke ◽  
Satoshi Murai
Keyword(s):  
Lower Bound ◽  
Simplicial Polytopes

scholarly journals Almost simplicial polytopes: the lower and upper bound theorems

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Eran Nevo ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon ◽  
David Yost
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Upper Bound Theorem ◽  
Simplicial Polytopes ◽  
Full Version ◽  
The Face ◽  
International Audience ◽  
Bound Theorem

International audience this is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of 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 s = 0. We characterize the minimizers and provide examples of maximizers, for any d.

scholarly journals Almost Simplicial Polytopes: The Lower and Upper Bound Theorems

2019 ◽  
Vol 72 (2) ◽  
pp. 537-556
Author(s):  
Eran Nevo ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon ◽  
David Yost
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Independent Interest ◽  
Upper Bound Theorem ◽  
Moment Curve ◽  
Simplicial Polytopes ◽  
Cyclic Polytopes ◽  
Bound Theorem ◽  
The Moment

AbstractWe study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called 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 the Upper Bound Theorem by McMullen, which treat the case where $s=0$. 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.

A generalized lower‐bound conjecture for simplicial polytopes

Mathematika ◽  
1971 ◽  
Vol 18 (2) ◽  
pp. 264-273 ◽  
Author(s):  
P. McMullen ◽  
D. W. Walkup
Keyword(s):  
Lower Bound ◽  
Simplicial Polytopes

scholarly journals Polytopes and $C^1$-convex bodies

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Karim Adiprasito ◽  
José Alejandro Samper
Keyword(s):  
Lower Bound ◽  
Convex Body ◽  
Hausdorff Distance ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Simplicial Polytopes ◽  
The Face ◽  
International Audience ◽  
Bound Theorem ◽  
Corps Convexe

International audience The face numbers of simplicial polytopes that approximate $C^1$-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence $\{P_n\}_{n=0}^{\infty}$ of simplicial polytopes converges to a $C^1$-convex body in the Hausdorff distance, then the entries of the $g$-vector of $P_n$ converge to infinity. Nous étudions les nombres de faces de polytopes simpliciaux qui se rapprochent de $C^1$-corps convexes dans la métrique Hausdorff. Plusieurs résultats structurels sur le skeleta de ces polytopes sont recherchées et utilisées pour calculer un théorème limite inférieure de cette classe de polytopes. Cela résout partiellement une conjecture formulée par Kalai en 1994: si une suite $\{P_n\}_{n=0}^{\infty}$ de polytopes simpliciaux converge vers une $C^1$-corps convexe dans la distance Hausdorff, puis les entrées du $g$-vecteur de $P_n$ convergent vers l’infini.

scholarly journals A Lower Bound Theorem for Centrally Symmetric Simplicial Polytopes

2018 ◽  
Vol 61 (3) ◽  
pp. 541-561 ◽  
Author(s):  
Steven Klee ◽  
Eran Nevo ◽  
Isabella Novik ◽  
Hailun Zheng
Keyword(s):  
Lower Bound ◽  
Simplicial Polytopes ◽  
Centrally Symmetric ◽  
Bound Theorem

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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