.  \[cor0\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body of the form . Let $m$ be the maximal number of linearly independent lattice vectors contained in ${\mathfrak{n}}\cap{\mathcal{B}}$. If $m<d$, then $$\label{eq0}\operatorname{vol}({\mathcal{B}})\ll_d|\Delta|^{\