On the mixing time of coordinate Hit-and-Run

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