# Reconstruct shape of a body from rationality of its projections

1. There is a closed convex body $S$ in $\mathbb{R}^3$. Areas of its projections on all planes (not only those normal to axes $x,y,z$) are rational numbers. Can we deduce that $S$ is a ball?
2. Replace rational with irrational in the first question.
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By continuity aren’t both hypotheses equivalent to the hypothesis that all projections have the same area? The answer to the two-dimensional analogue is no (curves of constant width), but I don’t know whether that generalizes to $\mathbb{R}^3$. –  Brian M. Scott Dec 8 '11 at 8:07