Peter McMullen

The maximum numbers of faces of a convex polytope

In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j d v , the maximum possible number of j -faces of a d -polytope with v vertices is achieved by a cyclic polytope C ( v, d ).

The numbers of faces of simplicial polytopes

In this paper is considered the problem of determining the possiblef-vectors of simplicial polytopes. A conjecture is made about the form of the sclution to this problem; it is proved in the case ofd-polytopes with at mostd+3 vertices.

Valuations on convex bodies

The investigation of functions on convex bodies which are valuations, or additive in Hadwiger’s sense, has always been of interest in particular parts of geometric convexity, and it has seen some progress in recent years. The occurrence of valuations in the theory of convex bodies can be traced back to the notion of volume in two essentially different ways. Firstly, the volume of convex bodies, being the restriction of a measure, is itself a valuation. This valuation property carries over to the functions which are deduced from volume in the Brunn-Minkowski theory, namely to mixed volumes, quermassintegrals, surface area functions, and others. Hadwiger’s celebrated characterizations of the quermassintegrals by the valuation and other properties were the culmination of a series of papers on valuations and at the same time the starting point for various subsequent investigations of functionals with similar properties.

On simple polytopes.

SummaryLetP be a simpled-polytope ind-dimensional euclidean space

Convex bodies which tile space by translation

\mathbb{E}^d

The polytope algebra

, and let Π(P) be the subalgebra of the polytope algebra Π generated by the classes of summands ofP. It is shown that the dimensions of the weight spacesΞr(P) of Π(P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality betweenΞr(P) andΞd-r(P). Moreover, Π(P) admits a Lefschetz decomposition under multiplication by the element ofΞ1(P) corresponding toP itself, which yields a proof of the necessity of McMullens conditions in theg-theorem on thef-vectors of simple polytopes. The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowskis second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities. A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them. The connexion found between Π(P) and the face ring of the dual simplicial polytopeP* enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP* is also established.

Lattice invariant valuations on rational polytopes

Publisher Summary This chapter focuses on valuations and dissections and discusses the state of knowledge of valuations and dissections. The concept of valuation lies at the heart of geometry, as does the closely related concept of a dissection. Fenchel–Jessen area measures can be obtained from a local version of the Steiner parallel formula. Variants on these measures are the Federer measures. Inner and outer angles can be used to find a relationship between polytope algebra and the polytope groups. The mixed valuations, derived from a continuous translation invariant or covariant valuation, are continuous in each of their arguments.

Lattice points in lattice polytopes

It is shown that a convex body K tiles E d by translation if, and only if, K is a centrally symmetric d -polytope with centrally symmetric facets, such that every belt of K (consisting of those of its facets which contain a translate of a given (d – 2)-face) has four or six facets. One consequence of the proof of this result is that, if K tiles E d by translation, then K admits a face-to-face, and hence a lattice tiling.