  In this paper we study the optimization problem
$$\mu^\star(\Omega):=\inf_{u\in \semi}\frac{\into |\n u|^2 \dx}{\into V u^2
\dx}$$ in a suitable functional space $\semi$. Here, $V$ is the multi-singular
potential given by $$V:=\sum_{1\leq i<j\leq n}
\frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}$$ and all the singular poles $a_1,
\ldots, a_n$, $n\geq 2$, arise either in the interior or at the boundary of a
smooth open domain $\Omega\subset \rr^N$, with $N\geq 3$ or $N \geq 2$,
respectively.
  For a bounded domain $\Omega$ containing all the singularities in the
interior, we prove that $\mu^\star(\Omega)>\mu^\star(\rr^N)$ when $n\geq 3$ and
$\mu^\star(\Omega)=\mu^\star(\rr^N)$ when $n=2$ (It is known from
\cite{cristi1} that $\mu^\star(\rr^N)=(N-2)^2/n^2)$.
  In the situation when all the poles are located on the boundary we show that
$\mu^\star(\Omega)=N^2/n^2$ if $\Omega$ is either a ball, the exterior of a
ball or a half-space. Our results do not depend on the distances between the
poles. In addition, in the case of boundary singularities we obtain that
$\mu^\star(\Omega)$ is attained in $\hoi$ when $\Omega$ is a ball and $n\geq
3$. Besides, $\mu^\star(\Omega)$ is attained in $\semi$ when $\Omega$ is the
exterior of a ball with $N\geq 3$ and $n\geq 3$ whereas in the case of a
half-space $\mu^\star(\Omega)$ is attained in $\semi$ when $n\geq 3$.
  We also analyze the critical constants in the so-called \textit{weak} Hardy
inequality which characterizes the range of $\mu's$ ensuring the existence of a
lower bound for the spectrum of the Schr\"{o}dinger operator $-\Delta -\mu V$.
In the context of both interior and boundary singularities we show that the
critical constants in the weak Hardy inequality are $(N-2)^2/(4n-4)$ and
$N^2/(4n-4)$, respectively.
