Minkowski bases, Korkin-Zolotarev bases and successive minima

Let $\lambda_k$ denote the $k$-th successive minimum of a lattice $L$. We study properties of the lengths of certain bases of $L$. If $v_1, \dots v_n$ is a basis which is reduced in the sense of Minkowski we show that $\lvert v_k \rvert^2 \leq \frac{k}{4} \lambda_{k}^2$ for $k = 6, 7$, confirming a conjecture of Sch\"urmann, and obtaining the first improvement of a classical bound by Van der Waerden. We construct a sequences of lattices where $\lvert v_n \rvert$ is significantly longer than the longest vector in a Korkin-Zolotarev reduced basis, answering a question of Sch\"urmann. In an appendix joint with Lior Hadassi we construct a lattice $L$ with the surprising property that any basis containing the shortest vector of $L$ is not the shortest basis.