Here’s a problem inspired by Luyi’s “Satiated Clam” problem. (Her problem makes me think of a clam with the biggest possible pearl inside.) The problem begins with the same setup as hers.
Let ABCDEF and ABC’D’E’F’ be two regular hexagons with unit side length. Suppose that their dihedral angle is . Let P be the convex hull of the two hexagons. For which is the volume of P maximal and what is this maximal volume?
I guess this one can be called the “Greedy Clam” problem, perhaps.
(Here’s a solution.)