Topic
Simple polytope
About: Simple polytope is a research topic. Over the lifetime, 146 publications have been published within this topic receiving 1744 citations.
Papers published on a yearly basis
Papers
[...]
TL;DR: In this article, it was shown that the dimensions of the weight spaces of a simpled-polytope are the h-numbers of the Dehn-Sommerville equations between the numbers of faces of the simplicial polytopeP and the classes of summands ofP, and that Π(P) admits a Lefschetz decomposition under multiplication by the element corresponding to P itself, which yields a proof of McMullen's conditions in the g-theorem on the f-vectors of simple polytopes.
Abstract: LetP be a simpled-polytope ind-dimensional euclidean space
$$\mathbb{E}^d $$
, and let Π(P) be the subalgebra of the polytope algebra Π generated by the classes of summands ofP It is shown that the dimensions of the weight spacesΞ
r(P) of Π(P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality betweenΞ
r
(P) andΞ
d-r
(P) Moreover, Π(P) admits a Lefschetz decomposition under multiplication by the element ofΞ
1(P) corresponding toP itself, which yields a proof of the necessity of McMullen's conditions in theg-theorem on thef-vectors of simple polytopes The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowski's second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them The connexion found between Π(P) and the face ring of the dual simplicial polytopeP
* enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP
* is also established
220 citations
1 Jan 2000
TL;DR: In this article, the complexity of Yamnitsky and Levin's Simplices Method for polytopes is discussed. But the complexity is not limited to simple polytope decomposition.
Abstract: Lectures on 0/l-Polytopes.- polymake: A Framework for Analyzing Convex Polytopes.- Flag Numbers and FLAGTOOL.- A Census of Flag-vectors of 4-Polytopes.- Extremal Properties of 0/1-Polytopes of Dimension 5.- Exact Volume Computation for Polytopes: A Practical Study.- Reconstructing a Simple Polytope from its Graph.- Reconstructing a Non-simple Polytope from its Graph.- A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm.- The Complexity of Yamnitsky and Levin's Simplices Method.
212 citations
TL;DR: In this paper, the authors studied the behavior of the f-and h-vector under barycentric subdivision of a simplicial complex and showed that the h-polynomial of its n-th iterated subdivision has convergent behavior.
Abstract: For a simplicial complex or more generally Boolean cell complex Δ we study the behavior of the f- and h-vector under barycentric subdivision. We show that if Δ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney–Davis conjecture for spheres that are the subdivision of a Boolean cell complex or the subdivision of the boundary complex of a simple polytope. For a general (d − 1)-dimensional simplicial complex Δ the h-polynomial of its n-th iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this h-polynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d.
96 citations
TL;DR: The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated in this article, where it is shown that the expected number of vertices is O(logdn) for any polytopes, the expected vertex count is Q(logd)-n) for a simple polytopes, and the expected vertices count is O (logd' n) for an infinite polytope.
Abstract: The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(logdn) for any polytope, the expected number of vertices is Q(logd-' n) for any simple polytope, and the expected number of facets is O(logd' n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time. EXTREME POINTS; GEOMETRIC PROBABILITY; AVERAGE-CASE ANALYSIS OF ALGORITHMS
68 citations
TL;DR: It is shown that for a certain class of numberings this algorithm has expected running time that is at worst quadratic in the dimension d, and a ‘successor-tuple’ for each vertex is defined which carries the crucial information of the numbering for local-improvement algorithms.
62 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 8 |
| 2020 | 6 |
| 2019 | 7 |
| 2018 | 9 |
| 2017 | 12 |
| 2016 | 6 |