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 $\lim_{n\to\infty} \log D_n/(n\log n)=2$. We also determine the exact value $D_4=4375/1024$.