Tverberg plus
constraints
Tverberg plus constraints
Florian Frick
TU Berlin
Drawing simplices
Florian Frick
TU Berlin
Tverberg’s
Theorem
Constraining
Tverberg’s
Theorem
Colorful Corollaries
3. July 2014
j/w Pavle Blagojević & Günter M. Ziegler
arXiv:1401.0690 and Bull. London Math. Soc. (2014)
Tetrahedron in the plane
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Tverberg plus
constraints
Tverberg’s Theorem
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Constraining
Tverberg’s
Theorem
Theorem (Tverberg 1966)
Let r , d ≥ 1 and N := (r − 1)(d + 1).
Rd
Then for any affine map f : ∆N →
there are r pairwise
disjoint faces σ1 , . . . , σr of ∆N with f (σ1 ) ∩ · · · ∩ f (σr ) 6= ∅.
Such a collection of faces σ1 , . . . , σr is called Tverberg
partition for f .
Colorful Corollaries
Topological Tverberg Theorem
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Theorem (Bárány, Shlosman & Szűcs 1981 – Özaydin
1987)
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Let r , d ≥ 1, where r is a power of a prime, and
N := (r − 1)(d + 1).
Then for any continuous map f : ∆N → Rd there are r
pairwise disjoint faces σ1 , . . . , σr of ∆N with
f (σ1 ) ∩ · · · ∩ f (σr ) 6= ∅.
Constraining Tverberg’s Theorem
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Lemma
Let r , d, c ≥ 1, where r is a power of a prime, and
N := (r − 1)(d + c + 1).
Rd
Then for any continuous maps f : ∆N →
and
c
g : ∆N → R there are r pairwise disjoint faces σ1 , . . . , σr of
∆N and x1 ∈ σ1 , . . . , xr ∈ σr with f (x1 ) = · · · = f (xr ) and
g (x1 ) = · · · = g (xr ).
Use g as constraint function.
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Constraining Tverberg’s Theorem
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Lemma
Let r , d, c ≥ 1, where r is a power of a prime, and
N := (r − 1)(d + c + 1).
Rd
Then for any continuous maps f : ∆N →
and
c
g : ∆N → R there are r pairwise disjoint faces σ1 , . . . , σr of
∆N and x1 ∈ σ1 , . . . , xr ∈ σr with f (x1 ) = · · · = f (xr ) and
g (x1 ) = · · · = g (xr ).
Use g as constraint function.
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Tverberg unavoidable subcomplexes
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Definition
Let r , d, N ≥ 1 and f : ∆N → Rd be a continuous map with
at least one Tverberg partition. Then a subcomplex Σ ⊆ ∆N
is called Tverberg unavoidable if for every Tverberg partition
σ1 , . . . , σr for f there is at least one face σi ⊆ Σ.
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Examples of unavoidable subcomplexes
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
1. Let S be a set of at most 2r − 1 vertices and Σ ⊂ ∆N
the subcomplex of faces with at most one vertex in S.
By pigeonhole principle at least one of r pairwise
disjoint faces has at most one vertex in S.
2. If k is an integer with r (k + 2) > N + 1 then the
(k)
k-skeleton ∆N is Tverberg unavoidable, since r
pairwise disjoint faces of dimension at least k + 1 would
require more than N + 1 vertices.
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Forcing partitions into unavoidable complexes
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Theorem
Tverberg’s
Theorem
Let r , d ≥ 1, where r is a power of a prime, and
N := (r − 1)(d + 2). Let Σ ⊆ ∆N be a Tverberg
unavoidable subcomplex.
Constraining
Tverberg’s
Theorem
Then for any continuous map f : ∆N → Rd there are r
pairwise disjoint faces σ1 , . . . , σr in Σ with
f (σ1 ) ∩ · · · ∩ f (σr ) 6= ∅.
Proof.
Consider the map ∆N → R, x 7→ dist(x, Σ).
Colorful Corollaries
Forcing partitions into unavoidable complexes
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Theorem
Tverberg’s
Theorem
Let r , d ≥ 1, where r is a power of a prime, and
N := (r − 1)(d + 2). Let Σ ⊆ ∆N be a Tverberg
unavoidable subcomplex.
Constraining
Tverberg’s
Theorem
Then for any continuous map f : ∆N → Rd there are r
pairwise disjoint faces σ1 , . . . , σr in Σ with
f (σ1 ) ∩ · · · ∩ f (σr ) 6= ∅.
Proof.
Consider the map ∆N → R, x 7→ dist(x, Σ).
Colorful Corollaries
Colorful Corollaries
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Corollary (Sarkaria 1991 – Volovikov 1996)
Let r be a power of a prime, d ≥ 1, N := (r − 1)(d + 2),
and k ≥ d r −1
r de.
Then for any continuous map f : ∆N → Rd there are r
pairwise disjoint faces σ1 , . . . , σr with
f (σ1 ) ∩ · · · ∩ f (σr ) 6= ∅ and dim σi ≤ k.
For r = 2 and d = 2 this is the non-planarity of K5 .
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Colorful Corollaries
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
This is a slightly stronger version of a theorem by Vrećica &
Živaljević (1994).
Corollary
Let r ≥ 2 be a prime power, d ≥ 1, c ≥ d r −1
r de + 1, and
N ≥ (r − 1)(d + 1 + c). Let f : ∆N → Rd be continuous,
and let the vertices of ∆N be divided into c color classes,
each of them of cardinality at most 2r − 1.
Then there are r pairwise disjoint rainbow faces σ1 , . . . , σr
of ∆N such that f (σ1 ) ∩ · · · ∩ f (σr ) 6= ∅.
For r = 2 and d = 2 this is the non-planarity of K3,3 .
Tverberg’s
Theorem
Constraining
Tverberg’s
Theorem
Colorful Corollaries
Tverberg plus
constraints
Florian Frick
TU Berlin
Drawing simplices
Tverberg’s
Theorem
Constraining
Tverberg’s
Theorem
Thank you!
Colorful Corollaries