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*+ 2*r*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!