  Recently, it has been shown by Balasubramanian et al. and Myers et al. that
the Bekenstein-Hawking entropy formula evaluated on certain closed surfaces in
the bulk of a holographic spacetime has an interpretation as the differential
entropy of a particular family of intervals (or strips) in the boundary theory.
We first extend this construction to bulk surfaces which vary in time. We then
give a general proof of the equality between the gravitational entropy and the
differential entropy. This proof applies to a broad class of holographic
backgrounds possessing a generalized planar symmetry and to certain classes of
higher-curvature theories of gravity. To apply this theorem, one can begin with
a bulk surface and determine the appropriate family of boundary intervals by
considering extremal surfaces tangent to the given surface in the bulk.
Alternatively, one can begin with a family of boundary intervals; as we show,
the differential entropy then equals the gravitational entropy of a bulk
surface that emerges from the intersection of the neighboring entanglement
wedges, in a continuum limit.
