  In this article we prove that for a smooth fiberwise convex Hamiltonian, the
asymptotic Hofer distance from the identity gives a strict upper bound to the
value at 0 of Mather's $\beta$ function, thus providing a negative answer to a
question asked by K. Siburg in \cite{Siburg1998}. However, we show that
equality holds if one considers the asymptotic distance defined in
\cite{Viterbo1992}.
