  In this paper we study centrally symmetric Birkhoff billiard tables. We
introduce a closed invariant set $\mathcal{M}_\mathcal{B}$ consisting of
locally maximizing orbits of the billiard map lying inside the region
$\mathcal{B}$ bounded by two invariant curves of $4$-periodic orbits. We give
an effective bound from above on the measure of this invariant set in terms of
the isoperimetric defect of the curve. The equality case occurs if and only if
the curve is a circle.
