Before revision:  Here we show that for convex bodies in R^{n} with diameter D, the resulting Coordinate Hit-and-Run (CHAR) algorithm mixes in poly(n ,D) steps. This is the first polynomial guarantee for this widely-used algorithm.
After revision:  Here we show that for convex bodies in R^{n} with diameter D, the resulting Coordinate Hit-and-Run (CHAR) algorithm on K is polynomial in n and D.