  We present an affine-invariant random walk for drawing uniform random samples
from a convex body $\mathcal{K} \subset \mathbb{R}^n$ that uses maximum volume
inscribed ellipsoids, known as John's ellipsoids, for the proposal
distribution. Our algorithm makes steps using uniform sampling from the John's
ellipsoid of the symmetrization of $\mathcal{K}$ at the current point. We show
that from a warm start, the random walk mixes in $\widetilde{O}(n^7)$ steps
where the log factors depend only on constants associated with the warm start
and desired total variation distance to uniformity. We also prove polynomial
mixing bounds starting from any fixed point $x$ such that for any chord $pq$ of
$\mathcal{K}$ containing $x$, $\left|\log \frac{|p-x|}{|q-x|}\right|$ is
bounded above by a polynomial in $n$.
