  The averaged null energy condition has known violations for quantum fields in
curved space, even if one considers only achronal geodesics. Many such examples
involve rapid variation in the stress-energy tensor in the vicinity of the
geodesic under consideration, giving rise to the possibility that averaging in
additional dimensions would yield a principle universally obeyed by quantum
fields. However, after discussing various procedures for additional averaging,
including integrating over all dimensions of the manifold, we give a class of
examples that violate any such averaged condition.
