Let $\mathbb{K}$ be a number field, let $r_1, r_2$ be the numbers of real resp. pairs of complex conjugate embeddings, and let $\sigma_i; \,i = 1,2, \ldots, r_1$ count the real embeddings, while $(\sigma_{r_1+i}, \sigma_{r_1+r_2+1}); \,i = 1, 2, \ldots, r_2$ are the pairs of complex conjugate embeddings. The norm indices $n_i = 1$ for $1 \leq i \leq r_1$ and $n_i = 2$ for $r_1 < i \leq r_1 + r_2$.

The log-Minkowski map $\ell : \mathbb{K}^{\times} \rightarrow \mathbb{R}^{r_1+r_2}$ is given by \begin{eqnarray*} x \mapsto \left( \log( | \sigma_i(x) |^{n_i} \right)_{i=1}^{r_1+r_2} . \end{eqnarray*}

Let $E \subset \mathcal{O}^{\times}(\mathbb{K})$ be a $\mathbb{Z}$-free submodule of the units such that $\mathcal{O}^{\times}(\mathbb{K}) = E \cdot W$, with $W \subset \mathbb{K}$ the roots of unity. Let $\upsilon_{\mathbb{K}} = \inf\{ | \ell( \varepsilon ) | : \varepsilon \in E \}$ be the smallest euclidean length of an $\ell$-image of some unit in $E$.

What lower bound for $\upsilon_{\mathbb{K}}$ can one prove? The case when the field is neither totally real nore CM is of particular interest.