5,250 research outputs found
Finite convex geometries of circles
Let F be a finite set of circles in the plane. We point out that the usual
convex closure restricted to F yields a convex geometry, that is, a
combinatorial structure introduced by P. H Edelman in 1980 under the name
"anti-exchange closure system". We prove that if the circles are collinear and
they are arranged in a "concave way", then they determine a convex geometry of
convex dimension at most 2, and each finite convex geometry of convex dimension
at most 2 can be represented this way. The proof uses some recent results from
Lattice Theory, and some of the auxiliary statements on lattices or convex
geometries could be of separate interest. The paper is concluded with some open
problems.Comment: 22 pages, 7 figure
A better upper bound on the number of triangulations of a planar point set
We show that a point set of cardinality in the plane cannot be the vertex
set of more than straight-edge triangulations of its convex
hull. This improves the previous upper bound of .Comment: 6 pages, 1 figur
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-dimensional simple polyhedron whose face poset is isomorphic to the
poset of ``non-crossing marked graphs'' with vertex set \A, where a marked
graph is defined as a geometric graph together with a subset of its vertices.
The poset of non-crossing graphs on \A appears as the complement of the star
of a face in that polyhedron.
The polyhedron has a unique maximal bounded face, of dimension
where is the number of points of \A in the interior of \conv(\A). The
vertices of this polytope are all the pseudo-triangulations of \A, and the
edges are flips of two types: the traditional diagonal flips (in
pseudo-triangulations) and the removal or insertion of a single edge.
As a by-product of our construction we prove that all pseudo-triangulations
are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
Representing convex geometries by almost-circles
Finite convex geometries are combinatorial structures. It follows from a
recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set
of planar convex polygons such that with respect to geometric
convex hulls is a locally convex geometry and every finite convex geometry can
be represented by restricting the structure of to a finite subset in a
natural way. An \emph{almost-circle of accuracy} is a
differentiable convex simple closed curve in the plane having an inscribed
circle of radius and a circumscribed circle of radius such that
the ratio is at least . % Motivated by Richter and
Rogers' result, we construct a set such that (1) contains
all points of the plane as degenerate singleton circles and all of its
non-singleton members are differentiable convex simple closed planar curves;
(2) with respect to the geometric convex hull operator is a locally
convex geometry; (3) as opposed to , is closed with respect
to non-degenerate affine transformations; and (4) for every (small) positive
and for every finite convex geometry, there are continuum
many pairwise affine-disjoint finite subsets of such that each
consists of almost-circles of accuracy and the convex geometry
in question is represented by restricting the convex hull operator to . The
affine-disjointness of and means that, in addition to , even is disjoint from for every
non-degenerate affine transformation .Comment: 18 pages, 6 figure
On the Complexity of Polytope Isomorphism Problems
We show that the problem to decide whether two (convex) polytopes, given by
their vertex-facet incidences, are combinatorially isomorphic is graph
isomorphism complete, even for simple or simplicial polytopes. On the other
hand, we give a polynomial time algorithm for the combinatorial polytope
isomorphism problem in bounded dimensions. Furthermore, we derive that the
problems to decide whether two polytopes, given either by vertex or by facet
descriptions, are projectively or affinely isomorphic are graph isomorphism
hard.
The original version of the paper (June 2001, 11 pages) had the title ``On
the Complexity of Isomorphism Problems Related to Polytopes''. The main
difference between the current and the former version is a new polynomial time
algorithm for polytope isomorphism in bounded dimension that does not rely on
Luks polynomial time algorithm for checking two graphs of bounded valence for
isomorphism. Furthermore, the treatment of geometric isomorphism problems was
extended.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the
Complexity of Isomorphism Problems Related to Polytopes'' (June 2001
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure
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