MathDB

A set of

n

points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets

\mathcal{A}

and

\mathcal{B}

. An

\mathcal{AB}

-tree is a configuration of

n-1

segments, each of which has an endpoint in

\mathcal{A}

and an endpoint in

\mathcal{B}

, and such that no segments form a closed polyline. An

\mathcal{AB}

-tree is transformed into another as follows: choose three distinct segments

A_1B_1

B_1A_2

, and

A_2B_2

in the

\mathcal{AB}

-tree such that

A_1

is in

\mathcal{A}

and

|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|

, and remove the segment

A_1B_1

to replace it by the segment

A_1B_2

. Given any

\mathcal{AB}

-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

Problem Statement