Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in $R^2$

It was proved in cite{NS1} that obstacles $K$ in $R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.