Summary: In order to express a polyhedron as the Minkowski sum of a polytope and a polyhedral cone, T. S. Motzkin [Beiträge zur Theorie der linearen Ungleichungen. Basel: University of Basel (Diss.) (1936)] devised a homogenization technique that translates the polyhedron to a polyhedral cone in one higher dimension. Refining his technique, we present a conical representation of a set in the Euclidean space. Then, we use this representation to reach four main results: First, we establish a convex programming based framework for determining a maximal element – an element with the maximum number of positive components – of a non-negative convex set – a convex set in the non-negative Euclidean orthant. Second, we develop a linear programming problem for finding a relative interior point of a polyhedron. Third, we propose two procedures for identifying a strictly complementary solution in linear programming. Finally, we generalize Motzkin’s [loc. cit.] representation theorem for a class of closed convex sets in the Euclidean space.