  In this paper we state the weighted Hardy inequality \begin{equation*}
c\int_{{\mathbb R}^N}\sum_{i=1}^n \frac{\varphi^2 }{|x-a_i|^2}\, \mu(x)dx\le
\int_{{\mathbb R}^N} |\nabla\varphi|^2 \, \mu(x)dx +k
\int_{\mathbb{R}^N}\varphi^2 \, \mu(x)dx \end{equation*} for any $ \varphi$ in
a weighted Sobolev spaces, with $c\in]0,c_o[$ where $c_o=c_o(N,\mu)$ is the
optimal constant, $a_1,\dots,a_n\in \mathbb{R}^N$, $k$ is a constant depending
on $\mu$. We show the relation between $c$ and the closeness to the single
pole. To this aim we analyze in detail the difficulties to be overcome to get
the inequality.
