Shortest network in 3D

Imagine a set of points in three dimensions. You want to connect the points with the shortest possible network. Sometimes you can make the network shorter by adding extra points. (These extra points are called Steiner points.) How much can extra points help? The answer is worth $1,000.

Here’s an example. Suppose our points are the corners of a unit cube. You can connect these points with a network of length 7. If you add a point in the center of the cube and connect every point to the center, you get a network of length 4 √3 = 6.928. So in this case, adding an extra point made it possible to reduce the size of the minimal spanning network by about 1%. You could do better by adding more points.

What is the most you can reduce the length of the minimum spanning network in three dimensions by adding extra points? The question concerns all possible sets of points, not just a particular set like the example above. It is conjectured that the most you can save is about 21.6%. That is, for any set of points, the ratio of the length of the shortest network with extra points to that of the shortest network without extra points is bounded below by

\sqrt{ \frac{283-3\sqrt{21}}{700} + \frac{9\sqrt{11 - \sqrt{21}}\sqrt{2}}{140}}

In their new book Magical Mathematics, Persi Diaconis and Ron Graham say “We currently offer a thousand dollars for a proof (or disproof) that this ratio is the best possible.”

As unwieldy as the number above appears, it makes some sense. It looks like the square roots come from repeated applications of the Pythagorean theorem. Someone may be able to reverse engineer the example the conjecture is based on by using the form of the proposed lower bound.

(Diaconis and Graham say that the corresponding problem in two dimensions have been solved and the optimal ratio is √3 / 2. However, this paper says that the conjecture is still unresolved, contrary to popular belief.)