355 research outputs found
A short proof of rigidity of convex polytopes
We present a much simplified proof of Dehn's theorem on the infinitesimal
rigidity of convex polytopes. Our approach is based on the ideas of Trushkina
and Schramm.Comment: to appear in Siberian Journal of Mathematics; 5 pages 2 figure
The rigidity of infinite graphs
A rigidity theory is developed for the Euclidean and non-Euclidean placements
of countably infinite simple graphs in R^d with respect to the classical l^p
norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and
Henneberg combinatorial characterisations of generic infinitesimal rigidity for
finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation
of the rigidity of generic finite body-bar frameworks in d-dimensional
Euclidean space is generalised to the non-Euclidean l^p norms and to countably
infinite graphs. For all dimensions and norms it is shown that a generically
rigid countable simple graph is the direct limit of an inclusion tower of
finite graphs for which the inclusions satisfy a relative rigidity property.
For d>2 a countable graph which is rigid for generic placements in R^d may fail
the stronger property of sequential rigidity, while for d=2 the equivalence
with sequential rigidity is obtained from the generalised Laman
characterisations. Applications are given to the flexibility of non-Euclidean
convex polyhedra and to the infinitesimal and continuous rigidity of compact
infinitely-faceted simplicial polytopes.Comment: 51 page
Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces
We state that any constant curvature Riemannian metric with conical
singularities of constant sign curvature on a compact (orientable) surface
can be realized as a convex polyhedron in a Riemannian or Lorentzian)
space-form. Moreover such a polyhedron is unique, up to global isometries,
among convex polyhedra invariant under isometries acting on a totally umbilical
surface. This general statement falls apart into 10 different cases. The cases
when is the sphere are classical.Comment: Survey paper. No proof. 10 page
Expansive Motions and the Polytope of Pointed Pseudo-Triangulations
We introduce the polytope of pointed pseudo-triangulations of a point set in
the plane, defined as the polytope of infinitesimal expansive motions of the
points subject to certain constraints on the increase of their distances. Its
1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of
the point set and whose edges are flips of interior pseudo-triangulation edges.
For points in convex position we obtain a new realization of the
associahedron, i.e., a geometric representation of the set of triangulations of
an n-gon, or of the set of binary trees on n vertices, or of many other
combinatorial objects that are counted by the Catalan numbers. By considering
the 1-dimensional version of the polytope of constrained expansive motions we
obtain a second distinct realization of the associahedron as a perturbation of
the positive cell in a Coxeter arrangement.
Our methods produce as a by-product a new proof that every simple polygon or
polygonal arc in the plane has expansive motions, a key step in the proofs of
the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by
Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially
to the "Further remarks" in Section 5) + changed to final book format. This
version is to appear in "Discrete and Computational Geometry -- The
Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds),
series "Algorithms and Combinatorics", Springer Verlag, Berli
Remarks on the combinatorial intersection cohomology of fans
We review the theory of combinatorial intersection cohomology of fans
developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This
theory gives a substitute for the intersection cohomology of toric varieties
which has all the expected formal properties but makes sense even for
non-rational fans, which do not define a toric variety. As a result, a number
of interesting results on the toric and polynomials have been extended
from rational polytopes to general polytopes. We present explicit complexes
computing the combinatorial IH in degrees one and two; the degree two complex
gives the rigidity complex previously used by Kalai to study . We present
several new results which follow from these methods, as well as previously
unpublished proofs of Kalai that implies and
.Comment: 34 pages. Typos fixed; final version, to appear in Pure and Applied
Math Quarterl
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