Sections of simplices and cylinders: Volume formulas and estimates

We investigate sections of simplices and generalized cylinders. We are interested in the volume of sections of these bodies with affine subspaces and give formulas and estimates for these volumes. For the regular n-simplex we state a general formula to compute the volume of the intersection with some k-dimensional subspace. A formula for central hyperplane sections was given by S. Webb. He also showed that the hyperplane through the centroid containing n-1 vertices gives the maximal volume. We generalize the formula to arbitrary dimensional sections that do not necessarily have to contain the centroid. And we show that, for a prescribed small distance of a hyperplane to the centroid, still the hyperplane containing n-1 vertices is volume maximizing. The proof also yields a new and short argument for Webb's result. The minimal hyperplane section is conjectured to be the one parallel to a face. We show that this hyperplane section is indeed minimal for dimensions n=2,3,4 and that it is a local minimum in general. Using results by Brehm e.a. we compute the average hyperplane section volume. For k-dimensional sections we give an upper bound. Finally we modify our volume formula to compute the section volume of irregular simplices. As an application we show that in odd dimensions larger than 4 there exist irregular simplices whose maximal section is not a face. A generalized cylinder is the Cartesian product of a n-dimensional cube and a m-dimensional ball of radius r. We study the behavior of the hyperplane section volume depending on the radius of the cylinder. First we show for the 3-dimensional cylinder that always a truncated ellipse gives the maximal volume. This is done by elementary geometric considerations and calculus. For the generalized cylinder we use the Fourier transform to derive an explicit formula. Then we estimate this by Hölder's inequality. Finally it remains to prove an integral inequality that is similar to the inequality of K. Ball for the cube

Slices, slabs, and sections of the unit hypercube

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Polya's Ph.D. thesis, and we discuss several applications of these formulas.Comment: 11 pages; minor corrections to reference

Projective geometry of Wachspress coordinates

We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves< are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic

When Does a Mixture of Products Contain a Product of Mixtures?

We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table

Weak hyperbolicity of cube complexes and quasi-arboreal groups

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is quasi-isometric to a tree. This implies that a group $G$ acting properly and cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove asymptotic finite-dimensionality for finitely-generated groups acting on finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded asymptotic dimension. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of $\Gamma$.Comment: Corrections in Sections 2 and 4. Simplification in Section