Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction

Once punctured disks, non-convex polygons, and pointihedra

We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometric properties of these graphs and how they relate to one another. In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic). We also find bounds on the diameters of these graphs, sometimes using the strongly convex embeddings. Finally, we show how these graphs relate to different polytopes, namely type D associahedra and a family of secondary polytopes which we call pointihedra.Comment: 24 pages, 6 figure

Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (\lfloor \tfrac{d}{2} \rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons'' that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets arbitrarily close to 9.Comment: 20 pages, 5 figure

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