  We prove a lower bound on the sharp Poincar\'e-Sobolev embedding constants
for general open sets, in terms of their inradius. We consider the following
two situations: planar sets with given topology; open sets in any dimension,
under the restriction that points are not removable sets. In the first case, we
get an estimate which optimally depends on the topology of the sets, thus
generalizing a result by Croke, Osserman and Taylor, originally devised for the
first eigenvalue of the Dirichlet-Laplacian. We also consider some limit
situations, like the sharp Moser-Trudinger constant and the Cheeger constant.
As a a byproduct of our discussion, we also obtain a Buser--type inequality for
open subsets of the plane, with given topology. An interesting problem on the
sharp constant for this inequality is presented.
