25 research outputs found
A Lower Bound for the Simplexity of then-Cube via Hyperbolic Volumes
AbstractLet T(n) denote the number of n -simplices in a minimum cardinality decomposition of the n -cube into n -simplices. For nâ„ 1, we show that T(n) â„H(n), where H(n) is the ratio of the hyperbolic volume of the ideal cube to the ideal regular simplex. H(n) â„12·6n/2(n+ 1)ân+12n!. Also limnâân [H(n)]1/nâ 0.9281. Explicit bounds for T(n) are tabulated for n†10, and we mention some other results on hyperbolic volumes
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
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
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
Lower Bounds for Simplicial Covers and Triangulations of Cubes
We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their exterior faces
Extremal properties for dissections of convex 3-polytopes
A dissection of a convex d-polytope is a partition of the polytope into
d-simplices whose vertices are among the vertices of the polytope.
Triangulations are dissections that have the additional property that the set
of all its simplices forms a simplicial complex. The size of a dissection is
the number of d-simplices it contains. This paper compares triangulations of
maximal size with dissections of maximal size. We also exhibit lower and upper
bounds for the size of dissections of a 3-polytope and analyze extremal size
triangulations for specific non-simplicial polytopes: prisms, antiprisms,
Archimedean solids, and combinatorial d-cubes.Comment: 19 page
Lectures on 0/1-polytopes
These lectures on the combinatorics and geometry of 0/1-polytopes are meant
as an \emph{introduction} and \emph{invitation}. Rather than heading for an
extensive survey on 0/1-polytopes I present some interesting aspects of these
objects; all of them are related to some quite recent work and progress.
0/1-polytopes have a very simple definition and explicit descriptions; we can
enumerate and analyze small examples explicitly in the computer (e.g. using
{\tt polymake}). However, any intuition that is derived from the analysis of
examples in ``low dimensions'' will miss the true complexity of 0/1-polytopes.
Thus, in the following we will study several aspects of the complexity of
higher-dimensional 0/1-polytopes: the doubly-exponential number of
combinatorial types, the number of facets which can be huge, and the
coefficients of defining inequalities which sometimes turn out to be extremely
large. Some of the effects and results will be backed by proofs in the course
of these lectures; we will also be able to verify some of them on explicit
examples, which are accessible as a {\tt polymake} database.Comment: 45 pages, many figures; to appear in "Polytopes - Combinatorics and
Computation" (G. Kalai and G.M. Ziegler, eds.), DMV Seminars Series,
Birkh"auser Base