  Given a matrix $A \in \mathbb{R}^{m \times n}$ ($n$ vectors in $m$
dimensions), and a positive integer $k < n$, we consider the problem of
selecting $k$ column vectors from $A$ such that the volume of the
parallelepiped they define is maximum over all possible choices. We prove that
there exists $\delta<1$ and $c>0$ such that this problem is not approximable
within $2^{-ck}$ for $k = \delta n$, unless $P=NP$.
