  We prove a new inequality controlling the large deviations of the empirical
measure of a Markov chain. This inequality is based on the martingale used by
Donsker and Varadhan and the minimax theorem. It holds for convex sets and it
requires to take an infimum over the starting point. In the case of a compact
space, this inequality is a partial improvement of the large deviations
estimates of Donsker and Varadhan. In the case of a non compact space, we
condition on the event that the process visits $n$ times a compact subset of
the space and we still obtain a control on the exponential scale.
