proved in [@Sandier_Serfaty] using the result of Montgomery on minimal theta function [@Mont], we provide below an alternative proof.

For any Bravais lattice $\Lambda$ $$W(\Lambda)=ah(\Lambda)+b,$$ where $a>0$, $b\in{\mathbb R}$ and $h(\Lambda)$ is the height of the flat torus ${\mathbb C}/\Lambda$ (see [@OPS; @Chiu; @CoulLazzarini] for more details).

Indeed, Osgood, Phillips and Sarnak [@OPS Section 4, page 205] proved, for $\Lambda={\mathbb Z}\oplus \tau{\mathbb Z}$, $\tau=a+ib$, that $$h(\Lambda)=-\log(b|\eta(\tau)|^4)+C, \quad C\in {\mathbb R},$$ where $\eta$ is the Dedekind eta function[^3]. But from [@Sandier_Serfaty] we have $$W(\Lambda)=-\frac{1}{2}\log\left(\sqrt{2\pi b}|\eta(\tau)|^2\right)+C,$$ therefore $W(\Lambda)=ah(\Lambda)+b$.

Then from [@OPS Corollary 1(b)], the triangular lattice minimizes $h$ among Bravais lattices with fixed density, hence the same is true for $W$.

Equilibrium Problem in the Whole Plane {#EPWP}
======================================

In this section we recall results on existence, uniqueness and characterization of the equilibrium measure $\mu_V$ and we give the definition of the admissible potentials.

Equilibrium measure, Frostman inequalities and differentiation of $U^{\mu_V}$
-----------------------------------------------------------------------------

[([@Bloom:2014qy])]{.nodecor} Let $K\subset {\mathbb R}^2$ be a compact set and let $\mathcal{M}_1(K)$ be the family of probability measures supported on $K$. Then the **logarithmic potential** and the **logarithmic energy** of $\mu\in \mathcal{M}_1(K)$ are defined as $$U^{\mu}(x):=-\int_{K}\log|x-y|d\mu(y) \quad \text{and} \quad I_0(\mu):=-\iint_{K\times K} \log|x-y|d\mu(x)d\mu(y).$$ We say that $K$ is **log-polar** if 