Extended gradient of convex function and capital allocation

We pose the problem of extending the notion of gradient of a convex function in such a way that the extended gradient exists and unique for every convex function at every point. We prove that this problem has a unique solution satisfying some natural axioms. This “special” extended gradient happens to be the Steiner point of the subdifferential set. We use this theory to develop, for the first time in the literature, a set of axioms for gradient-based capital allocation with convex positive homogeneous risk measures, such that the capital allocation satisfying these axioms always exists and unique. This result also has applications in the theory of risk sharing and cooperative investment.