19 research outputs found
Asymptotically efficient triangulations of the d-cube
Let and be polytopes, the first of "low" dimension and the second of
"high" dimension. We show how to triangulate the product
efficiently (i.e., with few simplices) starting with a given triangulation of
. Our method has a computational part, where we need to compute an efficient
triangulation of , for a (small) natural number of our
choice. denotes the -simplex.
Our procedure can be applied to obtain (asymptotically) efficient
triangulations of the cube : We decompose , for
a small . Then we recursively assume we have obtained an efficient
triangulation of the second factor and use our method to triangulate the
product. The outcome is that using and , we can triangulate
with simplices, instead of the achievable
before.Comment: 19 pages, 6 figures. Only minor changes from previous versions, some
suggested by anonymous referees. Paper accepted in "Discrete and
Computational Geometry
Lattice Delone simplices with super-exponential volume
In this short note we give a construction of an infinite series of Delone
simplices whose relative volume grows super-exponentially with their dimension.
This dramatically improves the previous best lower bound, which was linear.Comment: 7 pages; v2: revised version improves our exponential lower bound to
a super-exponential on
Lower bounds for the simplexity of the n-cube
In this paper we prove a new asymptotic lower bound for the minimal number of
simplices in simplicial dissections of -dimensional cubes. In particular we
show that the number of simplices in dissections of -cubes without
additional vertices is at least .Comment: 10 page
Triangulations of and Tropical Oriented Matroids
Develin and Sturmfels showed that regular triangulations of can be thought as tropical polytopes. Tropical oriented
matroids were defined by Ardila and Develin, and were conjectured to be in
bijection with all subdivisions of . In this
paper, we show that any triangulation of
encodes a tropical oriented matroid. We also suggest a new class of
combinatorial objects that may describe all subdivisions of a bigger class of
polytopes.Comment: 11 pages and 3 figures. Any comment or feedback would be welcomed v2.
Our result is that triangulations of product of simplices is a tropical
oriented matroid. We are trying to extend this to all subdivisions. v3
Replaces the proof of Lemma 2.6 with a reference.. Proof of the matrix being
totally unimodular is now more detailed. Extended abstract will be submitted
to FPSAC '1
There are only two nonobtuse binary triangulations of the unit -cube
Triangulations of the cube into a minimal number of simplices without
additional vertices have been studied by several authors over the past decades.
For this so-called simplexity of the unit cube is now
known to be , respectively. In this paper, we study
triangulations of with simplices that only have nonobtuse dihedral
angles. A trivial example is the standard triangulation into simplices. In
this paper we show that, surprisingly, for each there is essentially
only one other nonobtuse triangulation of , and give its explicit
construction. The number of nonobtuse simplices in this triangulation is equal
to the smallest integer larger than .Comment: 17 pages, 7 figure
Dyck path triangulations and extendability
We introduce the Dyck path triangulation of the cartesian product of two
simplices . The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever , any triangulation of
extends to a unique triangulation of
. Moreover, with an explicit construction, we
prove that the bound is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.Comment: 15 pages, 14 figures. Comments very welcome
Dyck path triangulations and extendability (extended abstract)
International audienceWe introduce the Dyck path triangulation of the cartesian product of two simplices . The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever, any triangulations of extends to a unique triangulation of . Moreover, with an explicit construction, we prove that the bound is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.Nous introduisons la triangulation par chemins de Dyck du produit cartésien de deux simplexes . Les simplexes maximaux de cette triangulation sont donnés par des chemins de Dyck, et cette construction se généralise de façon naturelle pour produire des triangulations qui utilisent des chemins de Dyck rationnels. Notre étude de la triangulation par chemins de Dyck est motivée par des problèmes de prolongement de triangulations partielles de produits de deux simplexes. On montre que alors toute triangulation de se prolonge en une unique triangulation de . De plus, avec une construction explicite, nous montrons que la borne est optimale. Nous présentons aussi des interprétations de nos résultats dans le langage des matroïdes orientés tropicaux, qui sont analogues aux résultats classiques de la théorie des matroïdes orientés