A polyhedral cone in dimension d is the solution set of a system of homogeneous linear non-strict inequalities in d variables. Equivalently, this is the conical hull of finitely many vectors with d coordinates. Here, the base field is that of the real numbers.

Suppose that C is the conical hull of k vectors with d coordinates. Then C is given by the matrix V with k columns and d columns, whose columns are the k vectors. The dual cone of C is the polyhedral set in dimension d which is the solution set of the system of homogeneous linear inequalities, whose matrix is the transpose of the matrix V.

The polyhedral cone C in dimension d is called simplicial if it is generated by d linearly independent vectors. A simplicial decomposition of C is a finite set of simplicial cones so that the union of their interiors (in the Euclidean topology) is equal to the interior of C.

Note that a polyhedral cone C, as a polyhedral set, has a single vertex which is the origin. In practice, it is convenient to use the term polyhedral cone for the translation of a polyhedral cone in the formal sense defined above. With this abuse of terminology, a polyhedral cone is given by a point (its apex, or vertex) and a number of vectors (its generating rays, or simply rays).