Place four pennies on a table such that each touches at least one other.
For what arrangement will the area of the convex hull of this shape be maximized?
(In reply to About this problem.
In this problem I made one major assumption - the figure is symmetric. This reduced a two variable problem down to a one variable problem. I also have confidence that this is the maximum since there is little reason for the figure to be asymmetric; but that does not guarantee that an asymmetric figure is not maximal.
5 pennies would be a very interesting problem, definitely D4 at a minimum possibly D5. But I would probably make an initial assumption that not only was the 5 penny figure symmetrical, but also that all the pennies' centers form a cyclic polygon. In the 4 penny case isosceles trapezoids are always cyclic.
I conjecture that for N pennies that the maximum convex hull occurs when all the centers of the pennies form an N sided cyclic polygon.