Four-point Fermat location problems revisited. New proofs and extensions of old results

What is the point at which the sum of (Euclidean) distances to four fixed points in the plane is minimised? This extension of the celebrated location question of Fermat about three points was partially solved by Fagnano around 1750, giving the following simple geometric answer: when the fixed points form a convex quadrangle it is the intersection point of both diagonals; it is not known who first derived the other case: otherwise it is the fixed point in the triangle formed by the three other fixed points. We show that the first case extends and generalises to general metric spaces, while the second case extends to any planar norm, any ellipsoidal norm in higher dimensional spaces and to the sphere.