Euclidean lattices with a metric

Let $L_1, L_2 \subseteq \mathbb{R}^n$  be full-rank Euclidean lattices with generating matrices $A, B$ respectively, and let $d(A,B)< \epsilon$, for some fixed positive real $\epsilon$. Then do there exist $\gamma, \delta \in \mathbb{R}^{+}$ (depending on $A,B, \epsilon$) such that \[\gamma < \operatorname{sup}_{x \in \mathbb{R}^n} |Ax-Bx|< \delta?\]

One clue could be that $\gamma, \delta$ somehow depend on the smallest and largest (in absolute value) eigenvalues of $A,B$.




Definition of $d(A,B)$: Given (full rank) lattices $L_1 = (a_{ij}), L_2=(b_{ij}) \in \mathbb{R}^n$ with gen. matrices $A, B$ respectively, we define \[d(L_1,L_2)=\sqrt{\sum_{i=1}^n \sum_{j=1}^n}(a_{ij}-b_{ij})^2.\]