Recently i've been thinking about base reduction of lattices, and this constant similar to Hermites constant came up. Let $L$ be a lattice with basis $\mathbf{b}_{1},\ldots,\mathbf{b}_{n}$. We define the orthogonality defect of the basis as $$ \delta(\mathbf{b}_{1},\ldots,\mathbf{b}_{n}) := \dfrac{\|\mathbf{b}_{1}\|\cdot\ldots\cdot\|\mathbf{b}_{n}\|}{\|\mathbf{b}_{1}^{*}\|\cdot\ldots\cdot\|\mathbf{b}_{n}^{*}\|}. $$ where $\mathbf{b}_{1}^{*},\ldots,\mathbf{b}_{n}^{*}$ denotes the Gram-Schmidt ortogonalization of the basis. Then we can define the orthogonality defect of a lattice $L$ as $$ \delta(L):=\min_{\substack{\mathbf{b}_{1},\ldots,\mathbf{b}_{n}\\ \text{ basis of $L$}}}{\delta(\mathbf{b}_{1},\ldots,\mathbf{b}_{n})}. $$ What i'm interested in is the constant $$ \delta_{n}:= \sup\limits_{L\in\mathcal{L}_{n}}\delta(L) $$ where $\mathcal{L}_{n}$ denotes the set of all $n$ dimensional lattices. Is there a name for this? Anyone know anything about this?
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It is known that
$$ (\sqrt{n}/12)^n\leq \delta_n \leq n^n. $$ See Theorem 1.41 and Lemma 1.42 of Rothvoss' notes.
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$\begingroup$ Thanks! I only knew the upper bound. I'm mainly interested in calculating the exact values in low dimensions. I wonder if anyone have done that already... $\endgroup$ Commented Sep 11 at 9:04
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1$\begingroup$ The exact bound for HKZ-reduced basis (so not exactly what you want, but likely “close”) has been computed in small dimensions, see here. $\endgroup$ Commented Sep 11 at 17:14