Consider a set of $n$ points in the plane. Among all the connected graphs (trees) $T$ in the plane that have these $n$ points among their vertices, I am looking to find one such that the sum of its edge lengths is minimum. Note that this question is different from Minimum Spanning Tree as we allow $T$ to have vertices other than the given $n$ points. For example, if $n=3$, there are 2 possibilities: Let $A,B,C$ be the three given points. If one of the angles in the triangle $ABC$, say $ABC$ is bigger than $120^\circ$ then AB+BC is the minimizing tree. If all the angles are less than $120^\circ$, let $P$ be the point within the triangle $ABC$ such that the angles $APB$, $BPC$, $CPA$ are all $120^\circ$. Then the tree $T$ with the set of vertices $A,B,C,P$ and edges $AP,BP,CP$ is the length minimizing one.
Is this problem worked out before?
