The "Little Prince" problem, which I learned from Greg Kuperberg, is a geometric answer to your question.
Here is the problem: the Little Prince stands in (I do mean in, not on) the plane and wants to shape its planet from a given quantity of matter (of given density) in order to maximize the gravity he feels. The most efficient way to go is to shape the planet as a round disk.
The problem has a particular point, the position of the Little Prince, but turns out to have a symmetric solution. Note that the same problem in higher dimension does not have a symmetric solution.
Let me add two points that make this example all the more interesting: first, the results still stands if the Little Prince is also authorized to shape the space (rather the surface) he lives in, with the constraint that it should have nonpositive curvature and be simply connected: he should still make the planet a round flat disk. Second, if one takes a general domain and integrates the inequality between the felt gravity and the optimal gravity, one gets the isoperimetric inequality.