In his remarkable book [On the Sphere and Cylinder][1], where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of parallel planes cutting the surface depends *only* on the distance between the planes (see p. 625 of this paper of [King][2]). This fact, which has been dubbed [Archimedes hatbox theorem][3], is now a standard exercise in many calculus texts, and is even commemorated on [the back of the Fields medal][4] (if you look closely there, you can see the figure below in the background).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="https://i.sstatic.net/ZKPoU.png" width="160">

Conversely, [Blaschke][5] showed that the only convex surface with this slab area property is the sphere. Indeed it is a simple exercise in classical differential geometry to check that any smooth convex surface with the slab area property must have constant curvature (Let $A(h)$ be the area trapped between a tangent plane and a prallel plane at distance $h$, and compute the limit of $A(h)/h$ as $h\to 0$). But can one still characterize the sphere if we fix the distance between the planes:

>**Question:** Let $S\subset R^3$ be a convex surface with diameter $d$. Suppose that for some constant $0<h<d$, the area of the portion of $S$ trapped between every pair of parallel planes separated by the distance $h$ is constant, whenever both planes intersect $S$. Does it follow then that $S$ is a sphere?

A *convex surface* is the boundary of a compact convex set with interior points, and its *diameter* is the distance between its farthest points in the ambient space. Although this is known to be an open problem,  I am not aware if it has been explicitly mentioned anywhere.

**Addendum:** See [this question][6] for a closely related problem on plank invariant measures for convex planar's domains, and more references in this area.


  [1]: https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder
  [2]: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/King609-628.pdf
  [3]: http://mathworld.wolfram.com/ArchimedesHat-BoxTheorem.html
  [4]: http://www.mathunion.org/general/prizes/fields/details/
  [5]: https://link.springer.com/book/10.1007%2F978-3-642-47392-0
  [6]: https://mathoverflow.net/questions/283370/plank-invariant-measures-on-convex-bodies