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, as $n\to\infty$, \[ \log D_n=2n\log n-n\log\log n+O(n). \] We also determine the exact value $D_4=4375/1024$.