  We give a simple argument to show that if {\alpha} is an affine isometric
action of a product G x H of topological groups on a reflexive Banach space X
with linear part {\pi}, then either {\pi}(H) fixes a unit vector or {\alpha}|G
almost fixes a point on X. It follows that any affine isometric action of an
abelian group on a reflexive Banach space X, whose linear part fixes no unit
vectors, almost fixes points on X.
