384 research outputs found
10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope
AbstractUsing oriented matroids, and with the help of a computer, we have found a set of 10 points inR4 not projectively equivalent to the vertices of a convex polytope. This result confirms a conjecture of Larman [6] in dimension 4
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
Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes
This article exhibits a 4-dimensional combinatorial polytope that has no
antiprism, answering a question posed by Bernt Lindst\"om. As a consequence,
any realization of this combinatorial polytope has a face that it cannot rest
upon without toppling over. To this end, we provide a general method for
solving a broad class of realizability problems. Specifically, we show that for
any semialgebraic property that faces inherit, the given property holds for
some realization of every combinatorial polytope if and only if the property
holds from some projective copy of every polytope. The proof uses the following
result by Below. Given any polytope with vertices having algebraic coordinates,
there is a combinatorial "stamp" polytope with a specified face that is
projectively equivalent to the given polytope in all realizations. Here we
construct a new stamp polytope that is closely related to Richter-Gebert's
proof of universality for 4-dimensional polytopes, and we generalize several
tools from that proof
Six topics on inscribable polytopes
Inscribability of polytopes is a classic subject but also a lively research
area nowadays. We illustrate this with a selection of well-known results and
recent developments on six particular topics related to inscribable polytopes.
Along the way we collect a list of (new and old) open questions.Comment: 11 page
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
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