In his remarkable book On the Sphere and Cylinder, 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. This fact, which has been dubbed Archimedes hatbox theorem, is now a standard exercise in many calculus texts, and is even commemorated on the back of the Fields medal (if you look closely, you will see a sphere inside a cylinder in the background).

Conversely, Blaschke 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 $d$ 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.