6
$\begingroup$

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).

  • 2
    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.2012-08-07

1 Answers 1

2

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.

  • 0
    @HenningMakholm I stole [this image](https://dl.dropbox.com/u/29863189/Blaschke_body.png) from page 113 of the very interesting book [Geometric Tomography](http://books.google.com/books/about/Geometric_Tomography.html?id=hwcKEZNLEmUC) by Rochard Gardner.2012-08-12