Orthogonality Defects of HKZ-Reduced Bases

Let $D_n$ denote the supremum of the orthogonality defects over all $n$-dimensional Hermite–Korkine–Zolotarev (HKZ)-reduced bases. This paper shows that ${\lim }_{n\to \mathrm{\infty }}\log D_n\mathrm{/}(n\log n)=2$. We also determine the exact value $D_4=4375\mathrm{/}1024$.