189 research outputs found
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Some basic mathematical tools such as convex sets, polytopes and combinatorial topology are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers.
One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincare formula(Poincare, 1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a correct account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space.
It turns out that in order to achieve (even partially) the above goals, I found that it was necessary to include quite a bit of background material on convex sets, polytopes, polyhedra and projective spaces. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also had to include some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of \projective polyhedron based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Some basic mathematical tools such as convex sets, polytopes and
combinatorial topology, are used quite heavily in applied fields such as
geometric modeling, meshing, computer vision, medical imaging and robotics.
This report may be viewed as a tutorial and a set of notes on convex sets,
polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay
Triangulations. It is intended for a broad audience of mathematically inclined
readers. I have included a rather thorough treatment of the equivalence of
V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and
H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin
elimination method (a version of Gaussian elimination for inequalities) is
discussed in some detail. I also included some material on projective spaces,
projective maps and polar duality w.r.t. a nondegenerate quadric in order to
define a suitable notion of ``projective polyhedron'' based on cones. To the
best of our knowledge, this notion of projective polyhedron is new. We also
believe that some of our proofs establishing the equivalence of V-polyhedra and
H-polyhedra are new.Comment: 183 page
Splitting Polytopes
A split of a polytope is a (regular) subdivision with exactly two maximal
cells. It turns out that each weight function on the vertices of admits a
unique decomposition as a linear combination of weight functions corresponding
to the splits of (with a split prime remainder). This generalizes a result
of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite
metric spaces.
Introducing the concept of compatibility of splits gives rise to a finite
simplicial complex associated with any polytope , the split complex of .
Complete descriptions of the split complexes of all hypersimplices are
obtained. Moreover, it is shown that these complexes arise as subcomplexes of
the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change
Many projectively unique polytopes
We construct an infinite family of 4-polytopes whose realization spaces have
dimension smaller or equal to 96. This in particular settles a problem going
back to Legendre and Steinitz: whether and how the dimension of the realization
space of a polytope is determined/bounded by its f-vector.
From this, we derive an infinite family of combinatorially distinct
69-dimensional polytopes whose realization is unique up to projective
transformation. This answers a problem posed by Perles and Shephard in the
sixties. Moreover, our methods naturally lead to several interesting classes of
projectively unique polytopes, among them projectively unique polytopes
inscribed to the sphere.
The proofs rely on a novel construction technique for polytopes based on
solving Cauchy problems for discrete conjugate nets in S^d, a new
Alexandrov--van Heijenoort Theorem for manifolds with boundary and a
generalization of Lawrence's extension technique for point configurations.Comment: 44 pages, 18 figures; to appear in Invent. mat
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