  Extremal functions are exhibited in Poincar\'e trace inequalities for
functions of bounded variation in the unit ball ${\mathbb B}^n$ of the
$n$-dimensional Euclidean space ${\mathbb R}^n$. Trial functions are subject to
either a vanishing mean value condition, or a vanishing median condition in the
whole of ${\mathbb B}^n$, instead of just on $\partial {\mathbb B}^n$, as
customary. The extremals in question take a different form, depending on the
constraint imposed. In particular, under the latter constraint, unusually
shaped extremal functions appear. A key step in our approach is a
characterization of the sharp constant in the relevant trace inequalities in
any admissible domain $\Omega \subset {\mathbb R}^n$, in terms of an
isoperimetric inequality for subsets of $\Omega$.
