  We consider averaged null energy conditions (ANEC) for strongly coupled
quantum field theories in even (two and four) dimensional curved spacetimes by
applying the no-bulk-shortcut principle in the context of the AdS/CFT duality.
In the same context but in odd-dimensions, the present authors previously
derived a conformally invariant averaged null energy condition (CANEC), which
is a version of the ANEC with a certain weight function for conformal
invariance. In even-dimensions, however, one has to deal with gravitational
conformal anomalies, which make relevant formulas much more complicated than
the odd-dimensional case. In two-dimensions, we derive the ANEC by applying the
no-bulk-shortcut principle. In four-dimensions, we derive an inequality which
essentially provides the lower-bound for the ANEC with a weight function. For
this purpose, and also to get some geometric insights into gravitational
conformal anomalies, we express the stress-energy formulas in terms of
geometric quantities such as the expansions of boundary null geodesics and a
quasi-local mass of the boundary geometry. We argue when the lowest bound is
achieved and also discuss when the averaged value of the null energy can be
negative, considering a simple example of a spatially compact universe with
wormhole throat.
