Effective rigidity away from the boundary for centrally-symmetric billiards

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.