5,690 research outputs found
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
Realization spaces of 4-polytopes are universal
Let be a -dimensional polytope. The {\em realization space}
of~ is the space of all polytopes that are combinatorially
equivalent to~, modulo affine transformations. We report on work by the
first author, which shows that realization spaces of \mbox{4-dimensional}
polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic
set~ defined over~, there is a -polytope whose realization
space is ``stably equivalent'' to~. This implies that the realization space
of a -polytope can have the homotopy type of an arbitrary finite simplicial
complex, and that all algebraic numbers are needed to realize all -
polytopes. The proof is constructive. These results sharply contrast the
-dimensional case, where realization spaces are contractible and all
polytopes are realizable with integral coordinates (Steinitz's Theorem). No
similar universality result was previously known in any fixed dimension.Comment: 10 page
Symmetry Matters for Sizes of Extended Formulations
In 1991, Yannakakis (J. Comput. System Sci., 1991) proved that no symmetric
extended formulation for the matching polytope of the complete graph K_n with n
nodes has a number of variables and constraints that is bounded
subexponentially in n. Here, symmetric means that the formulation remains
invariant under all permutations of the nodes of K_n. It was also conjectured
in the paper mentioned above that "asymmetry does not help much," but no
corresponding result for general extended formulations has been found so far.
In this paper we show that for the polytopes associated with the matchings in
K_n with log(n) (rounded down) edges there are non-symmetric extended
formulations of polynomial size, while nevertheless no symmetric extended
formulations of polynomial size exist. We furthermore prove similar statements
for the polytopes associated with cycles of length log(n) (rounded down). Thus,
with respect to the question for smallest possible extended formulations, in
general symmetry requirements may matter a lot. Compared to the extended
abtract that has appeared in the Proceedings of IPCO XIV at Lausanne, this
paper does not only contain proofs that had been ommitted there, but it also
presents slightly generalized and sharpened lower bounds.Comment: 24 pages; incorporated referees' comments; to appear in: SIAM Journal
on Discrete Mathematic
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
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