### The formulation

Finally, let’s make a precise formulation of the Local Steiner problem. There are two types of vertices in a Steiner tree, the given points or *terminals*, and the Steiner points. The collection of edges emanating from a terminal is called a *terminal configuration*, and from a Steiner point a *Steiner configuration*. Clearly, if is a Steiner point in a SMT with incident edges , , then the star consisting of vertices and edges is a SMT of . (A similar statement can be formulated for terminals.)

Thus we only have to consider stars. Without loss of generality, the centre of the star may be taken to be the origin . We may also assume that all the edges incident to are of unit length: first scale the star so that all edges are larger than unit length; then it is clear that the star from to all points on the edges at distance 1 from must also be a SMT. Note that Steiner configurations form a subclass of terminal configurations. Indeed, if the star with edges is a SMT of , then it is also a SMT of .

**The local Steiner problem.** *Given a finite-dimensional normed space , characterize all collections of unit vectors
*

*that form a terminal configuration, i.e., such that the star from to the is a SMT of , or**that form a Steiner configuration, i.e., such that the star from to the is a SMT of .*

As an example, the solution in Euclidean space is the following:

* forms*

*a terminal configuration if, and only if, and all angles ;**a Steiner configuration if, and only if, and all angles .*

In other normed spaces, our study of this problem utilizes various interesting fields of mathematics, in particular

- Convexity: covering and illumination of convex bodies,
- Convex Analysis: the subdifferential calculus,
- Combinatorics: extremal finite set theory,
- Banach space theory and linear algebra: Cotype and 1-summing constants

Thus this problem is not only of interest in itself. Instead, the connections to different fields of geometry and analysis enhances its importance.

### Two guiding conjectures

Let [, resp.] denote the maximum degree of a terminal [Steiner point, resp.] in a SMT in the normed space , where the maximum is taken over all SMTs. These two values measure what may be called the local complexity of a SMT in . A natural question is to determine these values for a given space . This should be possible in principle once the local Steiner problem is solved for . Observe that, since Steiner configurations are also terminal configurations, .

**Morgan’s conjecture**

Frank Morgan (1992, 1998 ) made the following conjecture:

**Conjecture.** *For any -dimensional normed space , .*

It is not difficult to show that . Indeed, the star joining the origin to the $2latex ^d$ vertices of the unit ball is a SMT of the vertices. Thus the upper bound of $2^d$, if true, would be best possible. However, at least when , there are other norms also attaining (Alfaro et al.1991).

In 2000 I showed that this conjecture holds for , and I characterized all the 2-dimensional with .

**Cieslik’s conjecture**

Dietmar Cieslik (1990, 1998 ) made a conjecture analogous to ‘s conjecture for .

**Conjecture.** *For any -dimensional normed space , . Equality holds for the -dimensional space with unit ball .*

Cieslik (1990b) proved this conjecture for the case , where the unit ball is an affine regular hexagon. The unit ball of is affinely equivalent to the rhombic dodecahedron.

### References

- M. Alfaro, M. Conger, K. Hodges, A. Levy, R. Kochar, L. Kuklinski, Z. Mahmood, and K. von Haam,
*The structure of singularities in -minimizing networks in*, Pacific J. Math. 149 (1991), 201-210. - D. Cieslik,
*Knotengrade kürzester Bäume in endlich-dimensionalen Banachräumen*, Proceedings of the 7th Fischland Colloquium, II (Wustrow, 1988), no. 39, 1990, pp. 89-93. - D. Cieslik, The vertex-degrees of Steiner minimal trees in Minkowski planes, Topics in Combinatorics and Graph Theory (R. Bodendiek and R. Henn, eds.), Physica-Verlag, Heidelberg, 1990, pp. 201-206.
- D. Cieslik,
*Steiner minimal trees*, Nonconvex Optimization and its Applications, vol.23, Kluwer Academic Publishers, Dordrecht, 1998. - F. Morgan (2000),
*Geometric measure theory*, third ed., Academic Press Inc., San Diego, CA, 2000. - F. Morgan (1992), Minimal surfaces, crystals, shortest networks, and undergraduate research, Math. Intelligencer
**14**, 37-44.

- F. Morgan (2000),
*Geometric measure theory*, third ed., Academic Press Inc., San Diego, CA, 2000. - F. Morgan (1992), Minimal surfaces, crystals, shortest networks, and undergraduate research, Math. Intelligencer
**14**, 37-44.