Grassmann angles of convex polytopes

Convex Polytopes

The study of convex polytopes in Euclidean space of two and three dimensions is one of the oldest branches of mathematics. Yet many of the more interesting properties of polytopes have been discovered comparatively recently, and are still unknown to the majority of mathematicians. In this paper we shall survey the subject, mentioning some of the most recent results, and stating the more importa...

Basic Properties of Convex Polytopes

Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial optimization. In this chapter we try to give a short introduction, pr...

Combinatorial Aspects of Convex Polytopes

Inner diagonals of convex polytopes

An inner diagonal of a polytope P is a segment that joins two vertices of P and that lies, except for its ends, in P ’s relative interior. For 1 ≤ i ≤ d, an i-diagonal of a d-polytope P is a segment [x, y] whose ends are vertices of P and whose carrier in P (i.e., the smallest face of P that contains [x, y]) is of dimension i. The number of P ’s i-diagonals is denoted by δi(P ). Thus δ1(P ) is ...

Volumes of Convex Lattice Polytopes

We show by a direct construction that there are at least exp{cV (d−1)/(d+1)} convex lattice polytopes in R of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family Pd(r) (to be defined in the text) of convex lattice polytopes whose volumes are between 0 and r/d!. Namely ...