91 research outputs found
A Geometric Lower Bound Theorem
We resolve a conjecture of Kalai relating approximation theory of convex
bodies by simplicial polytopes to the face numbers and primitive Betti numbers
of these polytopes and their toric varieties. The proof uses higher notions of
chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds
on the g-numbers of the approximating polytopes are given, in terms of their
Hausdorff distance from the convex body.Comment: 26 pages, 6 figures, to appear in Geometric and Functional Analysi
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unpublished proofs of Kalai that implies and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
Rigidity and volume preserving deformation on degenerate simplices
Given a degenerate -simplex in a -dimensional space
(Euclidean, spherical or hyperbolic space, and ), for each , , Radon's theorem induces a partition of the set of -faces into two
subsets. We prove that if the vertices of the simplex vary smoothly in
for , and the volumes of -faces in one subset are constrained only to
decrease while in the other subset only to increase, then any sufficiently
small motion must preserve the volumes of all -faces; and this property
still holds in for if an invariant of
the degenerate simplex has the desired sign. This answers a question posed by
the author, and the proof relies on an invariant we discovered
for any -stress on a cell complex in . We introduce a
characteristic polynomial of the degenerate simplex by defining
, and prove that the roots
of are real for the Euclidean case. Some evidence suggests the same
conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr
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