  In a recent paper V. Vasyunin presented a proof of the reverse H\"older
inequality with sharp constants for the weights satisfying the usual
Muckenhoupt condition. In this paper we present the inverse, that is, we use
the Bellman function technique to find the sharp $A_p$ constants for weights in
a reverse-H\"older class on an interval; we also find the sharp constants for
the higher-integrability result of Gehring.
  Additionally, we find bounds for the $A_p$ constants of
reverse-H\"older-class weights defined on rectangles and on cubes in n
dimensions.
