 
Summary: Discrete Comput Geom 21:117130 (1999) Discrete & Computational
Geometry© 1999 SpringerVerlag New York Inc.
Piles of Cubes, Monotone Path Polytopes, and
Hyperplane Arrangements
C. A. Athanasiadis
Mathematical Sciences Research Institute,
1000 Centennial Drive, Berkeley, CA 94720, USA
athana@msri.org
Abstract. Monotone path polytopes arise as a special case of the construction of fiber
polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permu
tahedron, which is a monotone path polytope of the standard unit cube. The permutahedron
is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the
cones of certain deformations of the braid arrangement can be realized as monotone path
polytopes. The construction is an extension of that of the permutahedron and yields in
teresting connections between enumerative combinatorics of hyperplane arrangements and
geometry of monotone path polytopes.
1. Introduction
Fiber polytopes were introduced by Billera and Sturmfels [5] and were further studied
in [6]. The fiber polytope (P, Q) is a polytope naturally associated to any projection
of polytopes : P Q. It is defined as
