[The piece below used to lie around on my homepage since April 2007. I'm slowly closing down that site, as the party is over at Google's Page Creator.]

The picture above is something that came out of my work with Achill Schürmann. It is used to disprove a conjecture of Gary Lawlor and Frank Morgan. Each of the grey figures is a Reuleaux triangle. They are all translates of each other. The three sides of a Reuleaux triangle are circular arcs, with the centre of each circle at the opposite vertex of the triangle. The three red figures are obtained by rotating the group of three grey figures by 180 degrees. Note that each grey triangle overlaps each pink triangle. In my opinion it would be quite difficult, using only pencil and paper, to discover a convex figure such that three nonoverlapping translates of it will overlap with each of the three figures obtained by a 180 degree rotation. I found it impossible to do by hand, and had to do a digital drawing, even though I knew theoretically of its possibility. See my paper with Achill for more information.

By the way, the figure was originally programmed in Postscript. More about this later.

Borsuk’s conjecture is true in dimensions 2 and 3. In 1993 Kahn and Kalai proved that the conjecture is false if the dimension is sufficiently large. For a short explanation by Gil Kalai, click here. For a wonderful (very) short story on how their proof came about, click here. The lowest dimension for which it is currently known that the conjecture fails is n=298, due to Aicke Hinrichs and Christian Richter.

It is not so difficult to prove the 2-dimensional case. The 3-dimensional case, first proved by Eggleston, is somewhat messy. A short proof for finite sets in 3-space was found by Heppes and Revesz (1956). It goes as follows.

The number of diameter pairs in a set S of n points in 3-space is at most 2n-2. (A diameter pair of S is a pair of points in S whose distance equals the diameter of S.)

Therefore, the diameter graph of S has average degree at most 4-4/n. So we can always find a point incident to at most 3 diameters of S.

So we can greedily 4-colour the graph.

The colour classes give the partition!

Problem: Can this argument be modified easily to also work for infinite sets? For example, a solution to the following problem would imply Eggleston’s theorem:

Show that there exist such that the graph G of almost-diameter pairs on any finite set S has chromatic number at most 4.

Here we define G of almost-diameter pairs to be the graph on S with edges all pairs of points at distance at least times the diameter of S.

In general, it is not clear how to go from finite sets to general bounded sets. So it is “conceivable” for example that the conjecture holds for all finite sets in 4-dimensional space, but not for all bounded sets….

By the way, according to MathSciNet, Erdös has dozens of posthomous papers, the newest having appeared in 2008….

I guess this is off-topic, but I’m always forgetting how to do stuff in . Here are two useful things:

Braces with labels

Do this:

$\ell_p^d=\overbrace{\ell_p^k\oplus\dots\oplus\ell_p^k}^{m \text{ times}}\oplus\ell_p^r$

to get this:

With the obvious variation

$\ell_p^d=\underbrace{\ell_p^k\oplus\dots\oplus\ell_p^k}_{m \text{ times}}\oplus\ell_p^r$

to get this:

Breaking “n-dimensional”

In order to convince to break -dimensional correctly, use

$n$\nobreakdash-\hspace{0pt}dimensional

(Otherwise will not do anything).

Latex blogs

For more serious stuff, see the Blog on Latex Matters or the Blog on Latex Matters (yes, you’re seeing double). The WordPress FAQ is also useful for latexblogging.

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

1. Convexity: covering and illumination of convex bodies,
2. Convex Analysis: the subdifferential calculus,
3. Combinatorics: extremal finite set theory,
4. 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.

In part I I discussed the Steiner problem in the Euclidean plane. Here I give an overview of various normed spaces for which geometric Steiner minimal trees have been considered. This motivates the study of Steiner minimal trees in arbitrary (finite-dimensional) normed spaces.

A finite-dimensional normed space (or Minkowski space) is a finite-dimensional real vector space on which a norm has been defined, i.e., a function such that

• for all with equality only for ;
  
• for all and ; and
• for all .

Higher dimensional Euclidean spaces

Mimura (1933) and Jarník and Kössler (1934) were among the first to consider Steiner minimal trees in Euclidean spaces (see also Korte and Nesetril 2001 for more history). Recently, attemps have been made to use high-dimensional Euclidean spaces to model phylogenetic trees (Brazil et al. 2008).

-geometry

Motivated by application in engineering, Hanan (1966) considered Steiner minimal trees in the -plane, i.e., , where . This is the distance between two points if one is only allowed to use vertical and horizontal lines. This norm is the main norm used in VLSI design. Recently, more orientations have also been considered. For example, if we allow three orientations, each at with respect to the other, we obtain the norm in which the unit ball is a regular hexagon. With four orientations, each at degrees, we obtain the norm with unit ball a regular octagon. In general we may consider the so-called -geometry, where the unit ball is the regular polygon with sides inscribed in the Euclidean circle. For even more general considerations, see for example Yan et al. (1997).

norm

The $\ell_p$ norm is very important in analysis, and has been considered in the Location Science literature to model distances between cities (Brimberg et al. 1991, 1993, Drezner et al. 2002). Weng (2008) uses this norm in the study of phylogenetic trees. The norm generalizes both the Euclidean norm and the norm. For it is defined on as

,

and for $p=\infty$ as

.

Gradient-constrained norm

The next example comes from the underground mining industry (Brazil et al. 1998, 2001). Here the problem is to design a network of tunnels connecting a set of given underground locations where ore is concentrated. Because of limitations in the trucks used to haul the ore, the tunnels are not allowed to be too steep. Thus we constrain the gradient of each edge to be at most , say. Apart from this constraint, the distance is Euclidean.

This distance function has the following formula, which satisfies the norm axioms.

if , and if .

The unit ball is the Euclidean unit ball with the north and south poles sliced off.

The one-dimensional Plateau problem

One viewpoint, which originates in the theory of minimal surfaces, is to consider Steiner minimal trees as the lowest dimensional case of the general Plateau problem. This is the problem of finding a set of smallest measure that makes a given set more connected. The classical Plateau problem is to find a surface in of smallest area which closes a given Jordan arc in 3-space. Here the study of singularities is very important, as they form an obstacle in getting a mathematical grip on minimizers (Morgan 2000). Similarly, the Euclidean Steiner problem is to find a network of smallest length which connects a given set of points. These consideration lead Frank Morgan and his students to study the local structure of Steiner minimal trees for various norms. They were mostly interested in piecewise- or uniformly convex norms (i.e., the curvature of the boundary of the unit ball is bounded away from 0). See Morgan (1992) for an exposition.

In the next installment, I’ll introduce the local Steiner problem.

References

• Y. Mimura (1933), Über die Bogenlänge, Ergebnisse eines Mathematischen Kolloquiums (K. Menger, ed.), vol. 4, Teubner, Leipzig und
  Wien, pp.~20–22.
  
• V. Jarník and M. Kössler (1934), O minimalnich grafeth obeahujicich danijch bodu, Cas. Pest. Mat. Fys. 63, 223-235.
• B. Korte and J. Nesetril (2001), Vojtech Jarník’s work in combinatorial optimization, Discrete Math. 235, 1-17.
• M. Brazil, B.K. Nielsen, D.A. Thomas, and M. Zachariasen (2008), A novel approach to phylogenetic trees: d-dimensional geometric Steiner trees,
  Networks, to appear.
• M. Hanan (1966), On Steiner’s problem with rectilinear distance, SIAM J. Appl. Math. 14, 255-265.
• G. Y. Yan, A. Albrecht, G. H. F. Young, and C. K. Wong (1997), The Steiner tree problem in orientation metrics, J. Comput. System Sci. 55, 529-546.
• J. Brimberg and R. F. Love (1991), Estimating travel distances by the weighted norm, Nav. Res. Logist. 38, 241-259
• J. Brimberg and R. F. Love (1993), Global convergence of a generalized iterative procedure for the minisum location problem with distances, Oper. Res. 41, 1153-1163.
• Z. Drezner and G. O. Wesolowsky (2002), Sensitivity analysis to the value of of the distance Weber problem, Ann. Oper. Res. 111, 135-150.
• M. Brazil, J. H. Rubinstein, D. A. Thomas, J. F. Weng, and N. C. Wormald (2001), Gradient-constrained minimum networks. I. Fundamentals, J. Global Optim. 21, 139-155.
• 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.

What is the shortest network that interconnects the three vertices of an equilateral triangle of edge length 1? Any pair of edges of the triangle form a minimal spanning tree of the three points. This tree has total length 2, but it is not the shortest.

Join the three vertices to the centroid of the triangle to obtain a tree of length .

This new point is the so-called Fermat-Torriceli point of the three vertices: the unique point whose sum of distances to the 3 vertices is a minimum. This tree is indeed the shortest tree joining the original 3 vertices, but it has one more vertex. The new vertex  is called a Steiner point, and the tree a Steiner minimal tree: a tree that is shortest among all those whose vertex set includes the original 3 points.

Consider the 4 vertices of a square of edge length 1. A minimal spanning tree has length 3.

As before, the network that joins all 4 vertices to the centre of the square is shorter: .

Indeed, the centre is again the Fermat-Torriceli point of the four given points. However, there is an even shorter tree interconnecting the 4 vertices, one with two Steiner points.

It has length , and is a Steiner minimal tree of the 4 vertices of the square. This example incidentally shows that Steiner minimal trees are in general not unique: rotating by gives another Steiner minimal tree.

In general, given any finite set of terminals in some space, a Steiner tree is any tree whose vertex set is a subset of the space and contains . A Steiner minimal tree (SMT) of is then any shortest Steiner tree of .

The Steiner problem can now be described as the problem of finding a SMT of any given finite set of points in the space. Much has been said about the history of this problem.

In particular we mention that Karl Menger already considered Steiner trees in general metric spaces in 1931, that Gilbert and Pollak (1968 introduced the name Steiner minimal tree, since Courant and Robbins (1941) referred to Steiner in their description of this problem. For a further historical information see Bern and Graham 1988, Hwang, Richards and Winter 1992, Cieslik 1998, and  Korte and Nesetril 2001.

The algorithmic problem of finding a SMT of a given set of points is already NP-hard in the Euclidean plane (Garey, Graham and Johnson 1977), but there are polynomial approximation schemes (Arora 1998), as well as exact algorithms that are feasible at least for up to 2000 points (Warme, Winter and Zachariasen 2000).

In the next installment, I’ll introduce spaces other than the Euclidean plane.

References

• S. Arora (1998), Polynomial time approximation schemes for {E}uclidean traveling salesman and other geometric problems, J. ACM 45, 753–782.
• M.W. Bern and R.L. Graham (1988), The shortest-network problem, Scientific American, January 1988, 84–89.
• Dietmar Cieslik (1998), Steiner minimal trees, Nonconvex Optimization and its
  Applications, vol.~23, Kluwer Academic Publishers, Dordrecht.
• R. Courant and H. Robbins (1941), What is mathematics?, Oxford Univ. Press, Oxford.
• M. R. Garey, R. L. Graham, and D. S. Johnson (1977), The complexity of computing Steiner minimal trees, SIAM J. Appl. Math. 32, 835–839.
• E. N. Gilbert and H. O. Pollak (1968), Steiner minimal trees, SIAM J. Appl. Math. 16, 1–29.
• F. K. Hwang, D. S. Richards, and P. Winter (1992), The Steiner tree problem, Ann. Discrete Math., vol. 53, North-Holland, Amsterdam.
• Bernhard Korte and Jaroslav Nesetril (2001), Vojtech Jarnik’s work in combinatorial optimization, Discrete Math. 235, 1–17.
• Karl Menger (1931), Some applications of point-set methods, Ann. of Math. (2) 32, 739–760.
• D. M. Warme, P. Winter, and M. Zachariasen (2000), Exact Algorithms for Plane Steiner Tree Problems: A Computational Study. In: Advances in Steiner Trees  (D.-Z. Du, J.~M. Smith, and J.~H. Rubinstein, eds.), Kluwer Academic Publishers, Boston, pp.~81–116.