Constant width bodies
There are planar figures, like the circle, which maintain the same height in every direction. These figures are called constant width bodies. The simplest example of this is the Reuleaux triangle. A natural question is whether there are examples of 3-dimensional constant width bodies. These bodies have been known to exist for over a century, but there were no concrete examples beside two examples by Meissner and the obvious constant width bodies of revolution. Instead their existence was shown by non-constructive proofs. Recently L. Montejano and E. Roldan-Pensado from UNAM in Mexico discovered new constructive methods to produce constant width bodies.