This is the first in a series of papers that explores a class of polyhedra we call 2-lattice polyhedra. 2-Lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyh...

A convex polyhedron P is k-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of P are k-gon for some fixed value of k. Since 1968 it is an open problem to construct all equiprojective polyhedra. Recently, Hasan and Lubiw [8] have given a characterization of equiprojective polyhedra. Based on their characterization, in this paper we discover s...

A new class of isogonal polyhedra is considered. Polyhedra are constructed using a combination of reflections in several symmetry planes of a given symmetry group. The procedure is a generalization of a Wythoff construction used for building uniform polyhedra. Every valid combination of symmetry planes generates an entire family of isogonal polyhedra. There are seven families with tetrahedral s...

We give an upper bound on the number of vertices of PI, the integer hull of a polyhedron P, in terms of the dimension n of the space, the number m of inequalities required to describe P, and the size ~ of these inequalities. For fixed n the bound is O(mn~n-1). We also describe an algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1 qE in...

We present lower bounds for the coefficients of Ehrhart polynomials of convex lattice polytopes in terms of their volume. We also introduce two formulas for calculating the Ehrhart series of a kind of a ”weak” free sum of two lattice polytopes and of integral dilates of a polytope. As an application of these formulas we show that Hibi’s lower bound on the coefficients of the Ehrhart series is n...

M.C. Escher drew a few of his patterns on finite polyhedra, but not on infinite polyhedra. In this paper we show some Escher-inspired patterns on triply periodic polyhedra.

In this paper we present anew, efficient, output-sensitive method for computing the visibility map of a set of c-oriented polyhedra (polyhedra with their faces and edges in c orientations, for some constant c) as seen from a given viewpoint. For non-intersecting polyhedra with n edges in total, the algorithm runs in time 0« n + k) log n), where k is the complexity ofthe visibility map. The meth...

Computing control invariant sets is paramount in many applications. The families of commonly used for computations are ellipsoids and polyhedra. However, searching a set over the family conservative systems more complex than unconstrained linear time systems. Moreover, even if may be approximated arbitrarily closely by polyhedra, complexity polyhedra grow rapidly certain directions. An attracti...

Simple equations are derived that give the shapes of various convex polyhedra. The five regular polyhedra, called Platonic solids (the tetrahedron, hexahedron or cube, octahedron, dodecahedron and icosahedron), and polyhedra composed of crystallographically low-index planes are treated. The equations also give shapes that are nearly polyhedral with round edges, or intermediate shapes between a ...

In this paper, we consider two classes of polyhedra. One is the class of polyhedra of maximal volume with n vertices that are inscribed in the unit sphere of R. The other class is polyhedra of minimal volume with n vertices that are circumscribed about the unit sphere of R. We construct such polyhedra for n up to 30 by a computer aided search and discuss some of their properties.