MinVolumeA note on the minimal volume of almost cubic
parallelepipedsDaniele Micciancio
http://link.springer.de/link/service/journals/00454/
Discrete and Computational Geometry291133-138200210.1007/s00454-002-2825-1

We prove that the best way to reduce the volume of the
n-dimensional unit cube by a linear transformation that
maps each of the main vertices e_{i} to
a point within distance epsilon < sqrt{(1/n) -
(1/n^{2})} from e_{i} is
to shorten all edges by a factor (1 - epsilon). In
particular, the minimal volume of such an almost cubic
parallelepiped is (1- epsilon)^{n}. This
problem naturally arises in the construction of lattice based
one-way functions with worst-case/average-case connection.