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  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
up vote 5 down vote accepted

As Brian M. Scott observes the orthogonal projections must all have the same area. Solids where all the projections are of constant area are known as bodies of constant brightness, and non-spherical bodies of constant brightness are known. The first discovered is known as a Blaschke-Firey body and is something like the body of rotation of a Reuleaux triangle.

An out of date introduction is the article Shapes of the Future by Victor Klee.

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I got curious and did a little searching. References and a minimality result can be found in this paper by Paolo Gronchi. In this paper Ralph Howard shows that balls are the only convex bodies with constant brightness and constant width. – Brian M. Scott Dec 8 '11 at 17:29

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