On ψ-direct sums of Banach spaces and convexity

Let X-1, X-2, ..., X-N be Banach spaces and psi a continuous convex function with some appropriate conditions on a certain convex set in RN-1. Let (X(1)circle plusX(2)circle plus...circle plusX(N))(psi), be the direct sum of X-1, X-2, ...,X-N equipped with the norm associated with psi. We characterize the strict, uniform, and locally uniform convexity of (X-1 circle plus X-2 circle plus ... circle plus X-N)(psi) by means of the convex function psi. As an application these convexities are characterized for the l(p,q)-sum (X-1 circle plus X-2 circle plus ... circle plus X-N)(p,q) (1 < q less than or equal to p less than or equal to infinity, q < infinity), which includes the well-known facts for the l(p)-sum (X-1 circle plus X-2 circle plus ... circle plus X-N)(p) in the case p = q.