The Blaschke--Lebesgue problem 

The Blaschke--Lebesgue theorem asserts that the Reuleaux triangle encloses least area among all constant width shapes in the plane. The Blaschke--Lebesgue problem is to find a least volume constant width body in space. I will discuss this problem and explain why I think the two conjectured volume-minimizing shapes are indeed solutions.

Bio
https://rhynd.math.upenn.edu/