  Using a polylogarithmic identity, we express the values of $\zeta$ at odd
integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple
functions involving products of logarithms. We also prove a useful property of
those functions as some of their variables are raised to a power. In the case
$n=2$, we prove two closed-form expressions concerning related integrals.
Finally, another family of related one-dimensional integrals is studied.
