Let $f$ map a point in the plane to the sum of the distances to each element in a given set of points. Can $f$ have multiple local minima?
For example, there is only one relative minimum when the given set is the set of vertices of an equilateral triangle, since if you're at some point in the plane other than the center of the triangle, you can always get a lower sum of distances by moving closer to the center (I think).