  We consider a class of non-local functionals recently introduced by H.
Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung, which offers a novel way
to characterize functions with bounded variation.
  We give a positive answer to an open question related to these functionals in
the case of functions with bounded variation. Specifically, we prove that in
this case the liminf of these functionals can be estimated from below by a
linear combination in which the three terms that sum up to the total variation
(namely the total variation of the absolutely continuous part, of the jump part
and of the Cantor part) appear with different coefficients. We prove also that
this estimate is optimal in the case where the Cantor part vanishes, and we
compute the precise value of the limit in this specific scenario.
  In the proof we start by showing the results in dimension one by relying on
some measure theoretic arguments in order to identify sufficiently many
disjoint rectangles in which the difference quotient can be estimated, and then
we extend them to higher dimension by a classical sectioning argument.
