Polyhedral cone

A polyhedral cone is a convex cone with a finite number of extreme rays.

Definition

Ziegler defines a polyhedral cone cone(Y) as the set of all linear combinations with nonnegative coefficients of a finite set of vectors Y . In the case of the empty set, we define ${\text{cone}}(\emptyset )=\{\mathbf {0} \}$.

Equivalently, a polyhedral cone can be defined as the convex hull of the origin and a finite set of rays extending from the origin.

Grunbaum generalizes this definition to allow an apex other than the origin, notated ${\text{cone}}_{a}\,Y$ where a  is the apex. Otherwise, the definition is the same.