  Given an exact symplectic map $T$ of a cylinder with a generating function
$H$ satisfying the so-called negative twist condition, $H_{12}>0$, we study the
locally maximizing orbits of $T$, that is, configurations which are local
maxima of the action functional $\sum_n H(q_n,q_{n+1})$. We provide a necessary
and sufficient condition for a configuration to be locally maximizing. Using
it, we consider a situation where $T$ has two generating functions with respect
to two different sets of symplectic coordinates. We suggest a simple geometric
condition which guarantees that the set of locally maximizing orbits with
respect to both of these generating functions coincide. As the main application
we show that the two generating functions for planar Birkhoff billiards satisfy
this geometric condition. We apply it to get the following result: consider a
centrally symmetric curve $\gamma$, for which the Birkhoff billiard map has a
rotational invariant curve $\alpha$ of $4$-periodic orbits. We prove that a
certain $L^2$-distance between $\gamma$ and its "best approximating" ellipse
can be bounded from above in terms of the measure of the complement of the set
filled by locally maximizing orbits lying between $\alpha$ and the boundary of
the phase cylinder. Moreover, this estimate is sharp, giving an effective
version of a recent result on Birkhoff conjecture for centrally symmetric
curves. We also get a similar bound for arbitrary curves $\gamma$, that relates
the measure of the complement of the set of locally maximizing orbits with the
$L^2$-distance between $\gamma$ and its "best approximating" circle.
