The center of a convex set and capital allocation

A capital allocation scheme for a company that has a random total profit Y and uses a coherent risk measure ρ has been suggested. The scheme returns a unique real number Λρ(X ,Y ), which determines the capital that should be allocated to company’s subsidiary with random profit X . The resulting capital allocation is linear and diversifying as defined by Kalkbrener (2005). The problem is reduced to selecting the “center” of a non-empty convex weakly compact subset of a Banach space, and the solution to the latter problem proposed by Lim (1981) has been used. Our scheme can also be applied to selecting the unique Pareto optimal allocation in a wide class of optimal risk sharing problems.