3,593 research outputs found
A Geometric Lower Bound Theorem
We resolve a conjecture of Kalai relating approximation theory of convex
bodies by simplicial polytopes to the face numbers and primitive Betti numbers
of these polytopes and their toric varieties. The proof uses higher notions of
chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds
on the g-numbers of the approximating polytopes are given, in terms of their
Hausdorff distance from the convex body.Comment: 26 pages, 6 figures, to appear in Geometric and Functional Analysi
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
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