  We consider the Dirichlet problem for positively homogeneous, degenerate
elliptic, concave (or convex) Hessian equations. Under natural and necessary
conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we
establish the interior $C^{1,1}$-regularity of the unique (admissible)
solution, which is optimal even if the boundary data is smooth. Both real and
complex cases are studied by the unified (Bellman equation) approach.
