97 research outputs found
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
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
The Gromov Norm of the Product of Two Surfaces
We make an estimation of the value of the Gromov norm of the Cartesian
product of two surfaces. Our method uses a connection between these norms and
the minimal size of triangulations of the products of two polygons. This allows
us to prove that the Gromov norm of this product is between 32 and 52 when both
factors have genus 2. The case of arbitrary genera is easy to deduce form this
one.Comment: The journal version contains an error that invalidates one direction
of the main theorem. The present version contains an erratum, at the end,
explaining thi
Enumerative Combinatorics
Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds
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