  The main contribution of this paper is that every convex function with
non-empty relative algebraic interior of its domain is Lipschitz and
subdifferentiable in some algebraic sense without any additional topological
constraints. The proposed approach uses slightly modified Clarke's
subdifferential for functions defined on a convex symmetric set and Lipschitz
with respect to a Minkowski functional. Following this, Clarke's
subdifferential calculus is generalized to vector spaces and, where continuity
properties are claimed, to topological vector spaces.
