  In this paper we give a geometric condition which ensures that
$(q,p)$-Poincar\'e-Sobolev inequalities are implied from generalized
$(1,1)$-Poincar\'e inequalities related to $L^1$ norms in the context of
product spaces. The concept of eccentricity plays a central role in the paper.
We provide several $(1,1)$-Poincar\'e type inequalities adapted to different
geometries and then show that our selfimproving method can be applied to obtain
special interesting Poincar\'e-Sobolev estimates. Among other results, we prove
that for each rectangle $R$ of the form $R=I_1\times I_2 \subset
\mathbb{R}^{n}$ where $I_1\subset \mathbb{R}^{n_1}$ and $I_2\subset
\mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have
that % \begin{equation*} \left( \frac{1}{w(R)}\int_{ R } |f
-f_{R}|^{p_{\delta,w}^*} \,wdx\right)^{\frac{1}{p_{\delta,w}^*}} \leq
c\,\delta^{\frac1p}(1-\delta)^{\frac1p}\,[w]_{A_{1,\mathfrak{R}}}^{\frac1p}\,
\Big(a_1(R)+a_2(R)\Big), \end{equation*} % where $\delta \in (0,1)$, $w \in
A_{1,\mathfrak{R}}$, $\frac{1}{p} -\frac{1}{ p_{\delta,w}^* }= \frac{\delta}{n}
\, \frac{1}{1+\log [w]_{A_{1,\mathfrak{R}}}}$ and $a_i(R)$ are bilinear analog
of the fractional Sobolev seminorms $[u]_{W^{\delta,p}(Q)}$ (See Theorem 2.18).
This is a biparameter weighted version of the celebrated fractional
Poincar\'e-Sobolev estimates with the gain
$\delta^{\frac1p}(1-\delta)^{\frac1p}$.
