  We introduce the concept of accessibility and prove that any convex body $X$
in $\mathbb R^d$ is accessible with relevant constants depending on $d$ only.
This property leads to a new algorithm which may be considered as a natural
derandomization of the hit and run algorithm applied to generate a sequence of
random points covering $X$ uniformly. We prove stability of the Markov chain
generated by the proposed algorithm and provide its rate of convergence.
