On the mixing time of coordinate Hit-and-Run

We obtain a polynomial upper bound on the mixing time TCHR(ε) of the coordinate Hit-and-Run random walk on an n-dimensional convex body, where TCHR(ε) is the number of steps needed in order to reach within ε 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 frac1ε, where we assume that the convex body contains the unit Vert·Vert_∞-unit ball B_∞ and is contained in its R-dilation R· B_∞. Whether coordinate Hit-and-Run has a polynomial mixing time has been an open question.