  We characterize the null energy condition for an $(n+1)$-dimensional
Lorentzian manifold in terms of convexity of the relative $(n-1)$-Renyi entropy
along displacement interpolations on null hypersurfaces. More generally, we
also consider Lorentzian manifolds with a smooth weight function and introduce
the Bakry-Emery $N$-null energy condition that we characterize in terms of null
displacement convexity of the relative $N$-Renyi entropy. As application we
then revisit Hawking's area monotonicity theorem for a black hole horizon and
the Penrose singularity theorem from the viewpoint of this characterization and
in the context of weighted Lorentzian manifolds.
