Let $V$ be an $n$-dimensional vector space and $L$ be a lattice in $V$, i.e. a free $\mathbb{Z}$-module of full rank. Define the following two numbers. Let $\lambda$ be the minimal positive real number such that the ball centered at the origin of radius $\lambda$ contains a basis of the lattice. Let $\mu$ be the minimal positive real number such that the ball centered at the origin of radius $\mu$ contains a basis of a sublattice of full rank. Apparantly $\lambda \geq \mu$. But it seems to me that (at least when $n=2$), these two numbers are always the same. Can one construct a example with $\lambda > \mu$?



a question on lattice invariants
Yes, as long as $n \geq 5$. The standard example is the lattice generated by
${\bf Z}^n$ and $(\frac12,\frac12,\ldots,\frac12)$. Then $\mu=1$ and 
$\lambda = n/4$. 


