991 research outputs found
Neighborly inscribed polytopes and Delaunay triangulations
We construct a large family of neighborly polytopes that can be realized with
all the vertices on the boundary of any smooth strictly convex body. In
particular, we show that there are superexponentially many combinatorially
distinct neighborly polytopes that admit realizations inscribed on the sphere.
These are the first examples of inscribable neighborly polytopes that are not
cyclic polytopes, and provide the current best lower bound for the number of
combinatorial types of inscribable polytopes (which coincides with the current
best lower bound for the number of combinatorial types of polytopes). Via
stereographic projections, this translates into a superexponential lower bound
for the number of combinatorial types of (neighborly) Delaunay triangulations.Comment: 15 pages, 2 figures. We extended our results to arbitrary smooth
strictly convex bodie
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
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
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