Tverberg Theorem (1965): Let be points in , . Then there is a partition of such that .
The (much easier) case of Tverberg’s theorem is Radon’s theorem.
1. Eckhoff’s Partition Conjecture
Eckhoff raised the possibility of finding a purely combinatorial proof of Tverberg’s theorem based on Radon’s theorem. He considered replacing the operation : “taking the convex hull of a set ” by an arbitrary closure operation.
Let be a set endowed with an abstract closure operation . The only requirements of the closure operation are:
(2) implies .
Define to be the largest size of a (multi)set in which cannot be partitioned into parts whose closures have a point in common.
Eckhoff’s Partition Conjecture: For every closure operation
If is the set of subsets of and is the convex hull operation then Radon’s theorem asserts that and Eckhoff’s partition conjecture would imply Tverberg’s theorem. Update (December 2010): Eckhoff’s partition conjecture was refuted by Boris Bukh. Here is the paper.
2. The dimension of Tverberg’s points
For a set , denote by those points in which belong to the convex hull of pairwise disjoint subsets of . We call these points Tverberg points of order .
Conjecture (Kalai, 1974): For every , .
Note that .
This conjecture includes Tverberg’s theorem as a special case: if , , and , then the sum in question is at most .
Akiva Kadari proved this conjecture (around 1980, unpublished) for planar configurations.
Akiva Kadari and Ziva Deutsch (both are my academic brothers).
3. The number of Tverberg’s partitions
Sierksma Conjecture: The number of Tverberg’s -partitions of a set of points in is at least .
4. The Topological Tverberg Conjecture
Let be a continuous function from the -dimensional simplex to . If then there are pairwise disjoint faces of whose images have a point in common.
If is a linear function this conjecture reduces to Tverberg’s theorem.
The case was proved by Bajmoczy and Barany using the Borsuk-Ulam theorem. In this case you can replace the simplex by any other polytope of the same dimension. (This can be asked also for the general case.)
The case where is a prime number was proved in a seminal paper of Barany, Shlosman and Szucs. The prime power case was proved by Ozaydin (unpublished), Volovikov, and Sarkaria. For the prime power case, the proofs are quite difficult and are based on computations of certain characteristic classes.
5. Reay’s Relaxed Tverberg Condition
Moriah Sigron (right) and other participants in a lecture by Endre Szemeredi. (See further comment below.)
Let be the smallest integer such that given points in , there exists a partition of such that every among the convex hulls , have a point in common.
Reay’s “relaxed Tverberg conjecture” asserts that that whenever , .
Micha A. Perles and Moriah Sigron have rather strong results in this direction, but at the same time Perles strongly believes that Reay’s conjecture is false, and he often mentions this special case:
Given 1,000,000 points in , Tverberg’s theorem asserts that you can partition them into 1,000 parts whose convex hulls have a point in common. Now given 999,999 points in is it always possible to divide them to 1,000 parts such that the convex hulls of every two of them will have a point in common? It is hard to believe that the answer is negative.
6. Colorful Tverberg theorems
Zivaljevic and Vrecica’s colorful Tverberg’s theorem asserts the following: Let be disjoint subsets of , called colors, each of cardinality at least . A -subset of is said to be multicolored if for . Let be an integer, and let denote the smallest value such that for every collection of colors of size at least there exist disjoint multicolored sets such that . Zivaljevic and Vrecica proved that for all , and if is a prime.
This theorem is one of the highlights of discrete geometry and topological combinatorics. The only known proofs for this theorem rely on topological arguments.
The colorful Tverberg conjecture asserts that .
Update: This conjecture was proved by Blagojecic Matschke and Ziegler
Let me mention another direction of moving from “colorful results” to analogous “matroidal results.” A set whose elements are colored with colors gives rise to a matroid where the rank of a set is the number of colors of elements in the set. So it is natural to consider an arbitrary matroid structure on the ground set and replace “multicolor set” by “a basis in the matroid”. For example, Barany’s colorful Caratheodory theorem was extended by Meshulam and me to a matroidal theorem. (With Barany and Meshulam we have some preliminary results on matroidal Tverberg theorems.)
7. The computational complexity of Tverberg’s theorem
Problem: Is there a polynomial-time algorithm to find a Tverberg partition when Tverberg’s theorem applies?
A positive answer will follow from a positive answer to:
Problem: Is there a polynomial algorithm for Barany’s colorful Caratheodory theorem?
The picture above (taken by Ofer Arbeli during a lecture by Endre Szemeredi at Hebrew University’s Institute for Advanced Study) shows how encouraging young babies (and even younger) to attend lectures, is instrumental in bringing Israeli mathematics and computer science to its leadership stature.