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Generalizing the well-known variety of plane curves of constant width, I'm wondering about three-dimensional surfaces of constant projected area.

Question: If $A$ is a (bounded) subset of $\mathbb R^3$, homeomorphic to a closed ball, such that the orthogonal projection of $A$ onto a plane has the same area for all planes, is $A$ necessarily a sphere? If not, what are some other possibilities?

Wikipedia mentions a concept of convex shapes with constant width, but that's different.

(Inspired by the discussion about spherical cows in comments to this answer -- my question is seeking to understand whether there are other shapes of cows that would work just as well).

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This is explicitly not what you want to know, but there's such a beautiful thread on MO about shapes with spherical projections mathoverflow.net/questions/39127 that it would be a pity not to mention it. –  t.b. Aug 7 '12 at 17:11
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up vote 2 down vote accepted

These are called bodies of constant brightness. A convex body that has both constant width and constant brightness is a Euclidean ball. But non-spherical convex bodies of constant brightness do exist; the first was found by Blaschke in 1916. See: Google and related MSE thread.

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Thanks. My google-fu is not strong enough to find an image of Blaschke's body (though there are many places referencing it). Do you know any? –  Henning Makholm Aug 10 '12 at 12:32
    
@HenningMakholm I stole this image from page 113 of the very interesting book Geometric Tomography by Rochard Gardner. –  user31373 Aug 12 '12 at 19:37
    
Thank you! ${}$ –  Henning Makholm Aug 12 '12 at 20:19
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