Fermat point in three-space

[Robert H. Lewis]

Hello,

  The Fermat point of a triangle ABC is the point, say P, that
minimizes the sum of the distances d(z,A) + d(z,B) + d(z,C) where z is
any point in the plane.  There was a good article on this in a recent
(May, I think) issue of the American Mathematical Monthly.  If all the
angles in ABC are less than 120 degrees, P is inside ABC.   Otherwise
it's one of the vertices.

   I couldn't find there any reference to the generalization to
3-space.  It seems there are two generalizations:  Given a tetrahedron
ABCD,

(1)  find the point that minimizes the sum  d(z,A) + d(z,B) + d(z,C) +
d(z,D).

(2) find the point that minimizes the sum of the areas of the
triangles formed by z and the six edges of ABCD (assume z is inside ABCD).

   Searching the web, I found only a comment (no proof) that soap films
assume the minimizing form on a wire frame tetrahedron.  That would be
problem (2).

  Any information on this problem would be of interest.

Robert H. Lewis
Mathematics
Fordham University
Bronx NY
rlewis at fordham.edu


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