V(x)}2 + \frac{\mathcal V(y)}2 \right)\,d\mu(x)\,d\mu(y)$$ among probability measures on $\mathcal S$ is achieved. Here $\|x-y\|$ denotes the euclidean norm in ${\mathbb R}^3$. Moreover, still from [@Hardy], the minimizer $\mu_{\mathcal V}$ is related to $\mu_V$ by the following relation $$\label{pullback}\mu_{\mathcal V} = T\sharp\mu_V,$$ where $T\sharp\mu$ denotes the push-forward of the measure $\mu$ by the map $T$.

\[ad\]

\[admissiblef\] We say that $V:{\mathbb R}^2\to {\mathbb R}$ is **admissible** if it is of class $C^3$ and if, defining $\mathcal V$ as above,

1.  [**(H1)**]{.nodecor} : The set $\{x\in {\mathbb R}^2 ; V(x)<+\infty\}$ is not log-polar and $\displaystyle \liminf_{|x|\to +\infty}\{V(x)-\log(1+|x|^2)  \}>-\infty.$

2.  [**(H2)**]{.nodecor} : The equilibrium measure $\mu_{\mathcal V}$ is of the form $m_{\mathcal V}(x)\indic_{\Sigma_{{\mathcal V}}}(x) \,dx$, where $m_{\mathcal V}$ is a $C^1$ function on $\mathcal S$ and $dx$ denotes the surface element on $\mathcal S$, where the function $m_{\mathcal V}$ is bounded above and below by positive constants $\bm$ and $\mb$, and where $\Sigma_{{{\mathcal V}}}$ is a compact subset of ${{\mathcal S}}$ with $C^1$ boundary.

Using (H2) and , we find that $$d\mu_V(x) = m_V(x) \indic_{\Sigma_V} \,dx,$$ where $\Sigma_V = T^{-1}(\Sigma_{{\mathcal V}})$ and $$\label{mv}m_V(x) = \frac{m_{\mathcal V}(T(x))}{(1+|x|^2)^2}.$$ Note that $(1+|x|^2)^{-2}$ is the jacobian of the transformation $T$.

Splitting Formula {#SpF}
=================

Assume $V$ 