2 Convex Polyhedron 3 2.1 What is convex polytope/polyhedron? . . . . . . . . . . . . . . . . . . . . . . 3 2.2 What are the faces of a convex polytope/polyhedron? . . . . . . . . . . . . . . 3 2.3 What is the face lattice of a convex polytope . . . . . . . . . . . . . . . . . . . 4 2.4 What is a dual of a convex polytope? . . . . . . . . . . . . . . . . . . . . . . . 4 2.5 What is simplex? . ....

Let S be a set of points in IR d , each with a weight that is not known precisely, only known to fall within some range. What is the locus of the centroid of S? We prove that this locus is a convex polytope, the projection of a zonotope in IR d+1. We derive complexity bounds and algorithms for the construction of these \centroid polytopes".

Given a graph G = (V,E) with node weights φv ∈ N ∪ {0}, v ∈ V , and some number F ∈ N∪{0}, the convex hull of the incidence vectors of all cuts δ(S), S ⊆ V with φ(S) ≤ F and φ(V \ S) ≤ F is called the bisection cut polytope. We study the facial structure of this polytope which shows up in many graph partitioning problems with applications in VLSI-design or frequency assignment. We give necessar...

For a reduced word i of the longest element in Weyl group SLn+1(C), one can associate string cone Ci which parametrizes dual canonical bases. In this paper, we classify all i's such that is simplicial. We also prove for any regular dominant weight λ sln+1(C), corresponding polytope Δi(λ) unimodularly equivalent to Gelfand–Cetlin associated if and only Thus completely characterize type polytopes...