#### Tverberg's Conjecture

Discrete Comput Geom
Geometry Discrete & Computational
Tverberg's Conjecture 0
Sinisˇa T. Vrec´ica 0
0 Faculty of Mathematics, University of Belgrade , Studentski trg 16, P.O.B. 550, 11000 Belgrade , Serbia
In 1989 Helge Tverberg proposed a quite general conjecture in Discrete Geometry, which could be considered as the common basis for many results in Combinatorial Geometry, and at the same time as a discrete analogue of the common transversal theorems. It implies or contains as special cases many classical “coincidence” results such as Radon's theorem, Rado's theorem, the Ham sandwich theorem, “non-embeddability” results (e.g. non-embeddability of graphs K5 and K3,3 in R2), etc. The main goal of this short note is to verify this conjecture in one new, non-trivial case. We obtain the continuous version of the conjecture. So, it is not surprising that we use topological methods, or more precisely the methods of equivariant topology and the theory of characteristic classes. ∗ This research was supported by the Ministry for Science, Technology and Development of Serbia, Grant 1854.
1. Introduction
Establishing the relation between Rado’s theorem on general measure (see [
11
]) and
the Ham sandwich theorem, the following result is proved in [
21
] stating that these two
results belong to the same family.
Theorem 1.1. Let 0 ≤ k ≤ d − 1, and let µ 0, µ 1, . . . , µ k be σ -additive probability
measures on Rd . Then there is a k-flat F with the property that every closed halfspace
containing F, has µ i -measure at least 1/(d − k + 1) for all i, 0 ≤ i ≤ k.
Namely, this theorem reduces to Rado’s theorem in the case k = 0 and to the Ham
sandwich theorem in the case k = d − 1.
The proof of the above theorem uses the topological result claiming the non-existence
of a non-zero section of a certain vector bundle over the Grassmann manifold. Helge
Tverberg observed that the special case k = 0 (Rado’s theorem) follows easily from his
result in Combinatorial Geometry from [
13
].
Theorem 1.2. Let S be a set of (r − 1)(d + 1) + 1 points in Rd . Then one can split it
into subsets S1, S2, . . . , Sr so that
r
i=1
conv Si = ∅.
This observation motivated him to suppose that a general result should exist which
would generalize his result, Theorem 1.2, and at the same time it would imply
Theorem 1.1 in the same way as his result implied Rado’s theorem. So, he formulated the
following Tverberg’s conjecture:
Conjecture 1.3. Let 0 ≤ k ≤ d − 1 and let S0, S1, . . . , Sk be finite sets of points in
Rd , with |Si | = (ri − 1)(d − k + 1) + 1 for i = 0, 1, . . . , k. Then Si can be split into ri
sets, Si1, Si2, . . . , Siri , so that there is a k-flat F meeting all the sets conv Sij, 0 ≤ i ≤
k, 1 ≤ j ≤ ri .
It is easy to see that this conjecture implies Theorem 1.1 ([
14
], see also [
20
]) and
its special case k = 0 is Theorem 1.2. This conjecture unifies two important themes of
Combinatorial Geometry: Helly-type theorems (the special case r = 2 of Theorem 1.2
is the well-known Radon’s theorem), and the common transversal theorems. Moreover,
it was shown in [
20
] that this result could be considered as an example of the whole
family of results of “combinatorial geometry on vector bundles”, and that these results
would also generalize many coincidence results such as the non-embeddability of the
graphs having minor K5 or K3,3 in the plane, etc.
Both Theorem 1.2 and Conjecture 1.3 could be reformulated in terms of linear
mappings and then generalized to the continuous case. Let us consider, for each i =
0, 1, . . . , k, the Ni -dimensional simplex iNi with the vertices e0i, e1i, . . . , eiNi , where
Ni = (ri − 1)(d − k + 1), and also the linear mapping ϕi : iNi → Rd sending the
vertices of iNi to the points of the set Si . We notice that the convex hull of some subset
of Si is the ϕi -image of the appropriate face of the simplex iNi . If we replace the linear
mappings by the continuous mappings, we obtain the continuous version of Tverberg’s
conjecture.
Conjecture 1.4. Let 0 ≤ k ≤ d − 1 and let ϕi : iNi → Rd for i = 0, 1, . . . , k be
continuous mappings. Then there is an affine k-flat F which intersects the images of ri
pairwise disjoint faces of simplex i for each i = 0, 1, . . . , k.
The special case k = 0 of this conjecture generalizes Theorem 1.2 and is known as
the continuous Tverberg theorem. It is established for r = 2 by Bajmo´czy and Ba´ra´ny
in [
1
] and when r is an odd prime number by Ba´ra´ny et al. in [
3
]. When r is a power
of a prime number, this is done by O¨zaydin in [
10
] and later also by Volovikov in [
16
],
Sarkaria in [
12
] and de Longueville in [
6
].
In [
15
] this conjecture was verified in some special cases. It was proved that, besides
the already known case k = 0, the conjecture is true in the case k = d − 1, and in the
case k = 1 when r0 = 1 or r1 = 1 or r0 = r1 = 2. Also, a slightly weakened version
of the conjecture is proved in the case k = d − 2, obtained when 3ri points in Si were
considered instead of 3ri − 2 of them. This version still suffices to imply Theorem 1.1
in the case k = d − 2.
A much more general result verifying some cases of this conjecture was achieved by
Zˇivaljevic´ in [
20
], where he established Conjecture 1.3 when r0 = r1 = · · · = rk is an
odd prime number, and both d and k are odd integers.
The main result of this paper is establishing Conjecture 1.3 in the case r0 = r1 =
· · · = rk = 2 without any additional restriction.
The result in [
20
] is obtained by using parametrized, ideal-valued, cohomological
index theory. The method of this paper will again be the reduction of the result to the
non-existence of some equivariant mapping, or equivalently to the non-existence of the
section of certain sphere bundle. This will be proved by showing that the corresponding
Stiefel–Whitney class is non-trivial. For the theory of characteristic classes consult [
9
].
Although Conjecture 1.3 is a statement in Combinatorial Geometry, all the results
establishing some of its special cases, establish its more general, continuous version
(Conjecture 1.4) and use topological methods. For very nice accounts on the topological
methods in Combinatorics and Combinatorial Geometry see [
2
], [
4
], [
18
], [
17
], [
19
]
and [
8
].
2. The Result
As we already mentioned, the main goal of this paper is to establish Conjecture 1.3 in
the case r0 = r1 = · · · = rk = 2, i.e. to prove the following theorem.
Theorem 2.1. Let 0 ≤ k ≤ d − 1 and let S0, S1, . . . , Sk be finite sets, each of d − k + 2
points in Rd . Then every set Si can be split into two sets, Si1 and Si2, so that there is a
k-flat F meeting all the sets conv Sij, 0 ≤ i ≤ k, j ∈ {1, 2}.
More generally, we will prove the continuous version of this statement, i.e. the case
r0 = r1 = · · · = rk = 2 of Conjecture 1.4. We consider the family of (d − k +
1)dimensional simplices 0, 1, . . . , k , where i is the simplex spanned by the vertices
e0i, e1i, . . . , edi−k+1 for i = 0, 1, . . . , k.
Theorem 2.2. Let 0 ≤ k ≤ d − 1 and let ϕi : i → Rd for i = 0, 1, . . . , k be
continuous mappings. Then there is an affine k-flat F which intersects the images of two
disjoint faces of every simplex i .
Proof. We denote with Grd,d−k the Grassmann manifold of (d − k)-dimensional linear
subspaces of Rd , and, for any L ∈ Grd,d−k , with πL : Rd → L the orthogonal projection.
We consider the (d − k + 1)-dimensional simplices i = conv{e0i, e1i, . . . , edi−k+1},
i = 0, 1, . . . , k. With ( i )2∗ we denote the deleted square of the simplex i , i.e. the set
of ordered pairs of points in i having disjoint supports. (A support of a point is a face
of the simplex containing it in its interior.) It is easy to verify that the deleted square
( i )2∗ is a (d − k)-dimensional manifold which we denote by M. It could be proved that
M is actually homeomorphic to the sphere Sd−k , but we do not need this fact here. (It is
shown in [
3
] that M is (d − k − 1)-connected.)
The statement of the theorem reduces to the claim that there exists L ∈ Grd,d−k such
that
πL (ϕ0(x0)) = πL (ϕ0(y0)) = · · · = πL (ϕk (xk )) = πL (ϕk (yk ))
for some ((x0, y0), . . . , (xk , yk )) ∈ ( 0)∗ × · · · × ( k )2∗. Here L = F ⊥, i.e. L is the
2
orthogonal complement to the affine k-flat F claimed to exist in the statement of the
theorem.
We denote with ξ the canonical vector bundle over Grd,d−k . For every L ∈ Grd,d−k ,
we have the mapping
ψL : ( 0)∗ × · · · × ( k )∗ → L2k+2,
2 2
ψL ((x0, y0), . . . , (xk , yk )) = (πL (ϕ0(x0)), πL (ϕ0(y0)), . . . , πL (ϕk (xk )), πL (ϕk (yk ))).
The group G = Z/2 ⊕ · · · Z/2 acts on these spaces, freely on ( 0)∗ × · · · × ( k )∗
2 2
≈ Mk+1 and fiberwise okn+1ξ 2k+2 (i.e. trivially on Grd,d−k and the i th generator of G
interchanges the (2i − 1)th and (2i )th coordinate of the fiber). The above mapping is
equivariant and it induces a section s of the vector bundle
ξ 2k+2 ×G Mk+1 → (Grd,d−k ×Mk+1)/G = Grd,d−k ×Mk+1/G.
The fiber over [L , (x0, y0, . . . , xk , yk )] could be identified with L2k+2. The statement
of the theorem now reduces to the claim that the section s intersects the diagonal in
some fiber. Let us suppose, to the contrary, that s does not intersect the diagonal in any
fiber. Projecting to the orthogonal complement of the diagonal and then radially to its
sphere (in each fiber), we obtain the non-zero section of the vector bundle with the fiber
L2k+1 and the section of the associated sphere bundle whose fiber is homeomorphic with
S(d−k)(2k+1)−1.
We reach a contradiction (proving in this way the theorem) by showing that the
topdimensional Stiefel–Whitney class of this sphere bundle does not vanish. The Poincare´
dual of the top-dimensional Stiefel–Whitney class of the sphere bundle coincides with
the homology class of the zero-set of a section of the associated vector bundle (with
the fiber L2k+1), which is transversal to the zero section. (For example, see [
7
] and
also Theorem 11.17 of [
5
], in the oriented case.) So, it suffices to find a section of the
corresponding vector bundle which intersects the zero section transversally in an odd
number of points.
In order to construct such a section we consider k + 1 parallel (d − k)-dimensional
affine planes A0, A1, . . . , Ak in Rd at distance m one from each other, and d −k +2 points
in each of them being the vertices v0i, v1i, . . . , vdi−k and the barycenter σˆi of a (d −
k)dimensional simplex σi , of diameter M, 0 ≤ i ≤ k. Let us suppose that they are in a
generic position meaning that their k + 1 barycenters span an affine k-dimensional flat F .
(Let us also choose them so that the flat F is orthogonal to the planes A0, A1, . . . , Ak .)
We also consider linear mappings fi which map vertices of i to the vertices and
the barycenter of σi , namely, fi (eij ) = vij , 0 ≤ j ≤ d − k and fi (edi−k+1) = σˆi for
i = 0, 1, . . . , k.
The images under the linear mapping fi of two disjoint faces of the simplex i
are the convex hulls of the corresponding vertices of the simplex σi and its
barycenter σˆi . The only non-empty intersection of the convex hulls of two disjoint subsets of
{v0i, v1i, . . . , vdi−k , σˆi } is
conv{v0i, v1i, . . . , vdi−k } ∩ conv{σˆi } = {σˆi }.
If some affine k-dimensional plane in Rd intersects the images of two disjoint faces
of some simplex i , then this plane contains the barycenter σˆi or it intersects Ai in at
least the one-dimensional affine plane. If this plane should intersect the images of two
disjoint faces of each simplex 0, 1, . . . , k , then (because of the generic position
of the planes A0, A1, . . . , Ak ) it has to contain the barycenters σˆ0, σˆ1, . . . , σˆk and it is
uniquely determined by them. So, the mappings f0, f1, . . . , fk induce the section of the
considered vector bundle which intersects the diagonal at a single orbit of the action of
the group G, i.e. at the orbit
L ,
0 0 k k
e0 + · · · + ed−k , ed0−k+1, . . . , e0 + · · · + ed−k , edk−k+1
d − k + 1 d − k + 1
where L = F ⊥ is the orthogonal complement to the k-dimensional flat F spanned by
the points σˆ0, σˆ1, . . . , σˆk .
So, we found a section which intersects the zero section of the considered vector
bundle in a single point. For small perturbation of the mappings fi , the intersection
with the zero section will remain a single point. Namely, it could be easily seen that for
continuous mappings gi sufficiently close to the mappings fi (let us say fi − gi < ε),
the only k-dimensional plane intersecting the gi -images of two disjoint faces of the
simplices i is again the plane spanned by the images of the barycenters, i.e. the points
g0(ed0−k+1), . . . , gk (edk−k+1). If some k-dimensional plane F did not contain the image
of some barycenter gi (edi−k+1), it would have to contain two different points (at the
distance bounded below by l − 2ε, where l is the minimal distance of the barycenter
of some simplex to its closest face) which are both very close to the plane Ai . This
would make the plane F contain a line (determined by those two points) very close
to the plane Ai (meaning that the angle α among that line and the plane Ai could be
as small as necessary, i.e. tan α ≤ 2ε/(l − 2ε)). Then the plane F could not intersect
the images of all simplices. It would be enough to choose ε to be much smaller than
(m · l)/4M . Namely, if F intersected the images of all simplices, then the angle β
among any line contained in F and the plane parallel with Ai would not be small, i.e.
tan β ≥ (m − 2ε)/(2M + 2ε), which is not true for the line considered above because
ε is chosen to be much smaller than (m · l)/4M . So, the obtained section intersects the
zero section transversally. This completes our proof.
Since our Theorem 2.2 establishes the continuous version of the conjecture in the case
r0 = r1 = · · · = rk = 2, it generalizes not only Tveberg’s theorem, but also the result of
Bajmo´ czy and Ba´ra´ny from [
1
] in the same way as the result from [
20
] generalizes the
result of Ba´ra´ny et al. from [
3
].
Our method does not use the fact that r = 2 is a prime, which might lead to the
hope that it could be used to establish the conjecture (or at least its special case k = 0)
for non-prime r . Unfortunately, it does not work for r > 2. Namely, in that case the
constructed section of the given vector bundle intersects the zero section in an even
number of points. Also, the configuration space corresponding to M k+1 in that case is
not a manifold, and so Poincare´ duality could not be used.
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