Slicing Convex Sets and Measures by a Hyperplane

Given convex bodies K ₁,…,K _d in ℝ ^d and numbers α ₁,…,α _d ∈[0,1], we give a sufficient condition for existence and uniqueness of an (oriented) halfspace H with Vol (H∩K _i )=α _i ⋅Vol K _i for every i. The result is extended from convex bodies to measures.     

Colorful polytopes and graphs

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P _G of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P _G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flagadjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds.     

Constructions for chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphismgroup has two orbits on flags, with adjacent flags lying indifferent orbits. In this paper, we describe a method for constructingfinite chiral n-polytopes, by seeking particular normal subgroupsof the orientation-preserving subgroup of an n-generator Coxetergroup (having the property that the subgroup is not normalizedby any reflection and is therefore not normal in the full Coxetergroup). This technique is used to identify the smallest examplesof chiral 3- and 4-polytopes, in both the self-dual and non-self-dualcases, and then to give the first known examples of finite chiral5-polytopes, again in both the self-dual and non-self-dual cases.