AbstractIn this paper we state the weighted Hardy inequality $$\begin{aligned} c\int _{{{\mathbb {R}}}^N}\sum _{i=1}^n \frac{\varphi ^2 }{|x-a_i|^2}\, \mu (x)dx\le \int _{{{\mathbb {R}}}^N} |\nabla \varphi |^2 \, \mu (x)dx +k \int _{\mathbb {R}^N}\varphi ^2 \, \mu (x)dx \end{aligned}$$ c ∫ R N ∑ i = 1 n φ 2 | x - a i | 2 μ ( x ) d x ≤ ∫ R N | ∇ φ | 2 μ ( x ) d x + k ∫ R N φ 2 μ ( x ) d x for any $$ \varphi $$ φ in a weighted Sobolev spaces, with $$c\in ]0,c_o[$$ c ∈ ] 0 , c o [ where $$c_o=c_o(N,\mu )$$ c o = c o ( N , μ ) is the optimal constant, $$a_1,\ldots ,a_n\in \mathbb {R}^N$$ a 1 , … , a n ∈ R N , k is a constant depending on $$\mu $$ μ . 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.