Eutactic lattice (awaiting review)

A lattice $L$ in $\mathbb R^n$ is called eutactic if there exist positive real numbers $c_1,\dots,c_m$ such that for every $v \in \mathbb R^n$, the inner product $$(v,v) = \sum_{j=1}^n c_j (v,x_j)^2,$$ where the sum is over all the minimal vectors of $L$. A eutactic lattice is well-rounded, and there are only finitely many similarity classes of eutactic lattices in each dimension $n \geq 2$. A lattice is called strongly eutactic if the eutaxy coefficients $c_1 = \dots = c_m$. Strongly eutactic lattices are arithmetic.