This problem is motivated by the Penny Boat Challenge: you are given an aluminum foil and you have to create a boat out of it that can hold the most amount of pennies.
I know that Archimedes' principle states that the upward buoyant force is equal to the weight of water displaced. This means that whatever shape is created, it should maximize the volume of water displaced for a fixed surface area.
Proven by the Isoperimetric inequality, a sphere is the most optimal closed surface for that maximizes volume per surface area. However, the shape that we create from the aluminum foil doesn't necessarily have to be closed (a boat can have an open top). So my question is, what is the optimal shape in this scenario?
I have not been able to find any rigorous mathematical analysis or proof on this subject, which is surprising because I feel like the answer to this question could have a lot of applications. Anyone know any good resources (or even a solution possibly) for this problem? Is this an open problem in mathematics?