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IMA Journal of Management Mathematics Advance Access originally published online on April 11, 2006
IMA Journal of Management Mathematics 2006 17(4):387-396; doi:10.1093/imaman/dpl007
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© The authors 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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

Frank Plastria**

MOSI Vrije Universiteit Brussel Pleinlaan 2, B 1050 Brussels, Belgium

** Email: frank.plastria{at}vub.ac.be

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.

Received January 2005. accepted on 16 December 2005.


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