123 research outputs found
Rolling of Coxeter polyhedra along mirrors
The topic of the paper are developments of -dimensional Coxeter polyhedra.
We show that the surface of such polyhedron admits a canonical cutting such
that each piece can be covered by a Coxeter -dimensional domain.Comment: 20pages, 15 figure
Volume estimates for equiangular hyperbolic Coxeter polyhedra
An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where
all dihedral angles are equal to \pi/n for some fixed integer n at least 2. It
is a consequence of Andreev's theorem that either n=3 and the polyhedron has
all ideal vertices or that n=2. Volume estimates are given for all equiangular
hyperbolic Coxeter polyhedra.Comment: 27 pages, 11 figures; corrected typo in Theorem 2.
In search for a perfect shape of polyhedra: Buffon transformation
For an arbitrary polygon consider a new one by joining the centres of
consecutive edges. Iteration of this procedure leads to a shape which is affine
equivalent to a regular polygon. This regularisation effect is usually ascribed
to Count Buffon (1707-1788). We discuss a natural analogue of this procedure
for 3-dimensional polyhedra, which leads to a new notion of affine -regular
polyhedra. The main result is the proof of existence of star-shaped affine
-regular polyhedra with prescribed combinatorial structure, under partial
symmetry and simpliciality assumptions. The proof is based on deep results from
spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro
Variational principles for circle patterns
A Delaunay cell decomposition of a surface with constant curvature gives rise
to a circle pattern, consisting of the circles which are circumscribed to the
facets. We treat the problem whether there exists a Delaunay cell decomposition
for a given (topological) cell decomposition and given intersection angles of
the circles, whether it is unique and how it may be constructed. Somewhat more
generally, we allow cone-like singularities in the centers and intersection
points of the circles. We prove existence and uniqueness theorems for the
solution of the circle pattern problem using a variational principle. The
functionals (one for the euclidean, one for the hyperbolic case) are convex
functions of the radii of the circles. The analogous functional for the
spherical case is not convex, hence this case is treated by stereographic
projection to the plane. From the existence and uniqueness of circle patterns
in the sphere, we derive a strengthened version of Steinitz' theorem on the
geometric realizability of abstract polyhedra.
We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and
Rivin for circle packings and circle patterns from our variational principles.
In the case of Br\"agger's and Rivin's functionals. Leibon's functional for
hyperbolic circle patterns cannot be derived directly from our functionals. But
we construct yet another functional from which both Leibon's and our
functionals can be derived.
We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics
Universality theorems for inscribed polytopes and Delaunay triangulations
We prove that every primary basic semialgebraic set is homotopy equivalent to
the set of inscribed realizations (up to M\"obius transformation) of a
polytope. If the semialgebraic set is moreover open, then, in addition, we
prove that (up to homotopy) it is a retract of the realization space of some
inscribed neighborly (and simplicial) polytope. We also show that all algebraic
extensions of are needed to coordinatize inscribed polytopes.
These statements show that inscribed polytopes exhibit the Mn\"ev universality
phenomenon.
Via stereographic projections, these theorems have a direct translation to
universality theorems for Delaunay subdivisions. In particular, our results
imply that the realizability problem for Delaunay triangulations is
polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure
On the convergence of the affine hull of the Chv\'atal-Gomory closures
Given an integral polyhedron P and a rational polyhedron Q living in the same
n-dimensional space and containing the same integer points as P, we investigate
how many iterations of the Chv\'atal-Gomory closure operator have to be
performed on Q to obtain a polyhedron contained in the affine hull of P. We
show that if P contains an integer point in its relative interior, then such a
number of iterations can be bounded by a function depending only on n. On the
other hand, we prove that if P is not full-dimensional and does not contain any
integer point in its relative interior, then no finite bound on the number of
iterations exists.Comment: 13 pages, 2 figures - the introduction has been extended and an extra
chapter has been adde
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