  We obtain a polynomial upper bound on the mixing time $T_{CHR}(\epsilon)$ of
the coordinate Hit-and-Run random walk on an $n-$dimensional convex body, where
$T_{CHR}(\epsilon)$ is the number of steps needed in order to reach within
$\epsilon$ of the uniform distribution with respect to the total variation
distance, starting from a warm start (i.e., a distribution which has a density
with respect to the uniform distribution on the convex body that is bounded
above by a constant). Our upper bound is polynomial in $n, R$ and
$\frac{1}{\epsilon}$, where we assume that the convex body contains the unit
$\Vert\cdot\Vert_\infty$-unit ball $B_\infty$ and is contained in its
$R$-dilation $R\cdot B_\infty$. Whether coordinate Hit-and-Run has a polynomial
mixing time has been an open question.
