The following questions occurred to me. This is not research mathematics, just idle curiosity. Apologies if it is inappropriate.
Suppose you have a fixed volume V of maleable material, perhaps clay. The goal is to form it into a shape S (convex or nonconvex) that would roll down an inclined plane as fast as possible. The plane is tilted at θ with respect to the horizontal. The race track that quantifies "as fast as possible" is of length L. The shape S must be entirely behind a starting plane orthogonal to the inclined plane, and its race is finished when it is entirely ahead of a finishing plane, L distant, again orthogonal to the inclined plane. So if S is a disk of radius r, its center of gravity will have to travel a total distance of L + 2r to complete the race. S is released at rest, and rolls under gravity. Assume that the materials of S and of the inclined plane have a sufficient coefficient of friction μ such that there is no slippage, just pure rolling.
The stipulation that S start behind and end ahead of the orthogonal planes suggests that the optimal shape is an arbitrarily thin (and therefore arbitrarily long) cylinder, to minimize r. But this does not accord with (my) physical intuition. What am I missing here?
A variation on the same problem replaces the inclined plane with an inclined half-cylinder of radius R, like a rain gutter. It may be necessary to assume a relation between R and the volume V of material to make this problem reasonable (so that S fits in the gutter). But certainly this version's answer could not be an arbitrarily long cylinder!