Shephard's Problem

Measurements of a centered convex body in Euclidean $n$-space (for $n\geq 3$) show that its brightness function (the volume of each projection) is smaller than that of another such body. Is it true that its Volume is also smaller? C. M. Petty and R. Schneider showed in 1967 that the answer is yes if the body with the larger brightness function is a projection body, but no in general for every $n$.

References

Gardner, R. J. ``Geometric Tomography.'' Not. Amer. Math. Soc. 42, 422-429, 1995.