Projection volumes of hyperplane arrangements

We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones is given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans that this held for all finite real reflection arrangements. The methods used are geometric and combinatorial. As a consequence we determine that the angle sums of a zonotope are given by the characteristic polynomial of the order dual of the intersection lattice of the arrangement

Fibre tilings

Generalizing an earlier notion of secondary polytopes, Billera and Sturmfels introduced the important concept of fibre polytopes, and showed how they were related to certain kinds of subdivision induced by the projection of one polytope onto another. There are two obvious ways in which this concept can be extended: first, to possibly unbounded polyhedra, and second, by making the definition a categorical one. In the course of these investigations, it became clear that the whole subject fitted even more naturally into the context of finite tilings which admit strong duals. In turn, this new approach provides more unified and perspicuous explanations of many previously known but apparently quite disparate results

Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n

We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice A_n in n-dimensional space are the identical rhombuses and equilateral triangles respectively.The prototiles obtained from orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root lattice of the Coxeter-Weyl group W(a)_n are classified. Orthogonal projections lead to various rhombuses and several triangles respectively some of which have been extensively discussed in the literature in different contexts. For example, rhombuses of the Voronoi cell of the root lattice A_4 projects onto only two prototiles: thick and thin rhombuses of the Penrose tilings. Similarly the Delone cells tiling the same root lattice projects onto two isosceles Robinson triangles which also lead to Penrose tilings with kites and darts. We point out that the Coxeter element of order h=n+1 and the dihedral subgroup of order 2n plays a crucial role for h-fold symmetric aperiodic tilings of the Coxeter plane. After setting the general scheme we give examples leading to tilings with 4-fold, 5-fold, 6-fold,7-fold, 8-fold and 12-fold symmetries with rhombic and triangular tilings of the plane which are useful in modelling the quasicrystallography with 5-fold, 8-fold and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A_(3)whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h=4 fold symmetry.Comment: 22 pages, 17 figure

Congruence and Metrical Invariants of Zonotopes

Zonotopes are studied from the point of view of central symmetry and how volumes of facets and the angles between them determine a zonotope uniquely. New proofs are given for theorems of Shephard and McMullen characterizing a zonotope by the central symmetry of faces of a fixed dimension. When a zonotope is regarded as the Minkowski sum of line segments determined by the columns of a defining matrix, the product of the transpose of that matrix and the matrix acts as a shape matrix containing information about the edges of the zonotope and the angles between them. Congruence between zonotopes is determined by equality of shape matrices. This condition is used, together with volume computations for zonotopes and their facets, to obtain results about rigidity and about the uniqueness of a zonotope given arbitrary normal-vector and facet-volume data. These provide direct proofs in the case of zonotopes of more general theorems of Alexandrov on the rigidity of convex polytopes, and Minkowski on the uniqueness of convex polytopes given certain normal-vector and facet-volume data. For a zonotope, this information is encoded in the next-to-highest exterior power of the defining matrix.Comment: 23 pages (typeface increased to 12pts). Errors corrected include proofs of 1.5, 3.5, and 3.8. Comments welcom

Convex Integer Optimization by Constantly Many Linear Counterparts

In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets $S\subset \Z^n$ of integer points presented by an oracle or by linear inequalities. Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of {\em edge complexity} of $S$, and use it to establish polynomial and constant upper bounds on the number of vertices of the projection \conv(WS) and on the number of linear optimization counterparts needed to solve the above convex problem. Two typical consequences are the following. First, for any $d$, there is a constant $m(d)$ such that the maximum number of vertices of the projection of any matroid $S\subset\{0,1\}^n$ by any binary $d\times n$ matrix $W$ is $m(d)$ regardless of $n$ and $S$; and the convex matroid problem reduces to $m(d)$ greedily solvable linear counterparts. In particular, $m(2)=8$. Second, for any $d,l,m$, there is a constant $t(d;l,m)$ such that the maximum number of vertices of the projection of any three-index $l\times m\times n$ transportation polytope for any $n$ by any binary $d\times(l\times m\times n)$ matrix $W$ is $t(d;l,m)$; and the convex three-index transportation problem reduces to $t(d;l,m)$ linear counterparts solvable in polynomial time