I'm currently working through "An Introduction to Mathematical Cryptography" by Hoffstein, Pipher and Silverman and I wonder how to cleanly define the term of "better" orthogonality of a basis $B$ of a lattice $L$.
The Hadamard-Ratio of the Basis $B = \{v_1,\dots,v_n\}$ is: $$\mathcal{H}(B) = \left(\frac{\det(L)}{||v_1||\cdots||v_n||}\right)^{1/n} $$ $B$ is strongly orthogonal if $\mathcal{H}(B) \approx 1$.
The term $\mathcal{H}(B) \approx 1$ is a bit vague since in an example $\mathcal{H}(B) \approx 0.7$ is also considered relatively orthogonal.
Is there a way to define the term of relative orthogonality more cleanly in terms of typical calculus a la $\varepsilon > 0$ and $\mathcal{H}(B) = 1- \varepsilon$ with $\varepsilon$ "small"?