In three-dimnesional space, you are asked to construct a polytope (a geometric object with flat sides) enclosing a volume of $1$. The cost of the model is the sum of the areas of the faces, plus some non-negative cost $\alpha$ for each edge (for each line joining two faces).
For a given $\alpha$, what is the best (minimal cost) such polytope?
If $\alpha = 0$ the problem has no solution, as you can make solids with flat sides arbitrarily close to a sphere. But for non-zero $\alpha$ such polytopes are prohibitively costly, as they have too many edges.
Try the problem for $\alpha = \frac{1}{40}$; it is not at all easy even for one special case. The general pattern as $\alpha$ decreases from some high value ($2$ is high enough!) will be a progression from a regular tetrahedron, to a different shape with more edges, to yet another shape, and so forth. The question is, what is the progression of shapes?