  The Ryu-Takayanagi formula relates entanglement entropy in a field theory to
the area of extremal surfaces anchored to the boundary of a dual AdS space. It
is interesting to ask if there is also an information theoretic interpretation
of the areas of non-extremal surfaces that are not necessarily
boundary-anchored. In general, the physics outside such surfaces is associated
to observers restricted to a time-strip in the dual boundary field theory. When
the latter is two-dimensional, it is known that the differential entropy
associated to the strip computes the length of the dual bulk curve, and has an
interpretation in terms of the information cost in Bell pairs of restoring
correlations inaccessible to observers in the strip. A general realization of
this formalism in higher dimensions is unknown. We first prove a no-go theorem
eliminating candidate expressions for higher dimensional differential entropy
based on entropic c-theorems. Then we propose a new formula in terms of an
integral of shape derivatives of the entanglement entropy of ball shaped
regions. Our proposal stems from the physical requirement that differential
entropy must be locally finite and conformally invariant. Demanding
cancellation of the well-known UV divergences of entanglement entropy in field
theory guides us to our conjecture, which we test for surfaces in $AdS_4$. Our
results suggest a candidate c-function for field theories in arbitrary
dimensions.
