Rigidity of Scattering Lengths and Traveling Times for Disjoint Unions of Convex Bodies

Obstacles $K$ and $L$ in $R^d$ ($d\ge 2$) are considered that are finite disjoint unions of strictly convex domains with $C^3$ boundaries. We show that if $K$ and $L$ have (almost) the same scattering length spectrum, or (almost) the same traveling times, then $K=L$.