  We introduce the concept of local Poincar\'e constant of a $BV$ function as a
tool to understand the relation between its mean oscillation and its total
variation at small scales. This enables us to study a variant of the BMO-type
seminorms on $\varepsilon$-size cubes introduced by Ambrosio, Bourgain, Brezis,
and Figalli. More precisely, we relax the size constraint by considering a
family of functionals that allow cubes of sidelength smaller than or equal to
$\varepsilon$. These new functionals converge, as $\varepsilon$ tends to zero,
to a local functional defined on $BV$, which can be represented by integration
in terms of the local Poincar\'e constant and the total variation. This
contrasts with the original functionals, whose limit is defined on $SBV$ and
may not exist for functions with a non-trivial Cantor part. Moreover, we
characterize the local Poincar\'e constant of a function with a cell-formula
given by the maximum mean oscillation of its $BV$ blow-ups. As a corollary of
this characterization, we show that the new limit functional extends the
original one to all $BV$ functions. Finally, we discuss rigidity properties and
other challenging questions relating the local Poincar\'e constant of a
function to its fine properties.
