Surfaces of constant projected area

Question

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).

This is explicitly not what you want to know, but there's such a beautiful thread on MO about shapes with spherical projections http://mathoverflow.net/questions/39127/ that it would be a pity not to mention it. — t.b., Aug 07 '12 at 17:11

score 2 · Accepted Answer · edited Apr 13 '17 at 12:20

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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.

edited Apr 13 '17 at 12:20

Community

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answered Aug 08 '12 at 18:05

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? – hmakholm left over Monica Aug 10 '12 at 12:32
@HenningMakholm I stole this image from page 113 of the very interesting book Geometric Tomography by Rochard Gardner. – Aug 12 '12 at 19:37

Surfaces of constant projected area

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