12,797 research outputs found
Axioms for the g-vector of general convex polytopes
McMullen's g-vector is important for simple convex polytopes. This paper
postulates axioms for its extension to general convex polytopes. It also
conjectures that, for each dimension d, a stated finite calculation gives the
formula for the extended g-vector. This calculation is done by computer for d=5
and the results analysed. The conjectures imply new linear inequalities on
convex polytope flag vectors. Underlying the axioms is a hypothesised
higher-order homology extension to middle perversity intersection homology
(order-zero homology), which measures the failure of lower-order homology to
have a ring structure.Comment: LaTeX2e. 10 page
On the metric dimension of rotationally-symmetric convex polytopes
Metric dimension is a~generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let be a family of connected graphs : depending on as follows: the order and . If there exists a constant C > 0 such that for every then we shall say that has bounded metric dimension, otherwise has unbounded metric dimension. If all graphs in have the same metric dimension, then is called a family of graphs with constant metric dimension.\\ In this paper, we study the metric dimension of some classes of convex polytopes which are rotationally-symmetric. It is shown that these classes of convex polytoes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension
Practical Volume Estimation by a New Annealing Schedule for Cooling Convex Bodies
We study the problem of estimating the volume of convex polytopes, focusing
on H- and V-polytopes, as well as zonotopes. Although a lot of effort is
devoted to practical algorithms for H-polytopes there is no such method for the
latter two representations. We propose a new, practical algorithm for all
representations, which is faster than existing methods. It relies on
Hit-and-Run sampling, and combines a new simulated annealing method with the
Multiphase Monte Carlo (MMC) approach. Our method introduces the following key
features to make it adaptive: (a) It defines a sequence of convex bodies in MMC
by introducing a new annealing schedule, whose length is shorter than in
previous methods with high probability, and the need of computing an enclosing
and an inscribed ball is removed; (b) It exploits statistical properties in
rejection-sampling and proposes a better empirical convergence criterion for
specifying each step; (c) For zonotopes, it may use a sequence of convex bodies
for MMC different than balls, where the chosen body adapts to the input. We
offer an open-source, optimized C++ implementation, and analyze its performance
to show that it outperforms state-of-the-art software for H-polytopes by
Cousins-Vempala (2016) and Emiris-Fisikopoulos (2018), while it undertakes
volume computations that were intractable until now, as it is the first
polynomial-time, practical method for V-polytopes and zonotopes that scales to
high dimensions (currently 100). We further focus on zonotopes, and
characterize them by their order (number of generators over dimension), because
this largely determines sampling complexity. We analyze a related application,
where we evaluate methods of zonotope approximation in engineering.Comment: 20 pages, 12 figures, 3 table
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
- …