Notes About the Carathéodory Number
Discrete Comput Geom
Notes About the Carathéodory Number
Imre Bárány 0 1 2 3
Roman Karasev 0 1 2 3
0 R. Karasev ( ) Dept. of Mathematics, Moscow Institute of Physics and Technology , Institutskiy per. 9, Dolgoprudny 141700 , Russia
1 I. Bárány Department of Mathematics, University College London , Gower Street, London WC1E 6BT, England
2 I. Bárány Rényi Institute of Mathematics, Hungarian Academy of Sciences , P.O. Box 127, 1364 Budapest , Hungary
3 R. Karasev Laboratory of Discrete and Computational Geometry, Yaroslavl State University , Sovetskaya st. 14, Yaroslavl 150000 , Russia
In this paper we give sufficient conditions for a compactum in Rn to have Carathéodory number less than n + 1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverbergtype theorem for families of convex compacta.
Carathéodory's theorem; Helly's theorem; Tverberg's theorem

1 Introduction
The Carathéodory theorem [
7
] (see also [
10
]) asserts that every point x in the convex
hull of a set X ⊂ Rn is in the convex hull of one of its subsets of cardinality at most
n + 1. In this note we give sufficient conditions for the Carathéodory number to be
less than n + 1 and prove some related results. In order to simplify the reasoning, we
always consider compact subsets of Rn.
There are results about lowering the Carathéodory constant: A theorem of
Fenchel [
11
] (see also [
12
]) asserts that a compactum X ⊂ Rn either has the
Carathéodory number ≤n or can be separated by a hyperplane into two nonempty
parts. By separated we mean “divided by a hyperplane disjoint from X into two
nonempty parts.” In order to state more results, we need formal definitions.
Definition 1.1 For a compactum X ⊂ Rn, we denote by convk X the set of points
p ∈ Rn that can be expressed as a convex combination of at most k points in X. We
denote by conv X (without subscript) the standard convex hull of X.
Definition 1.2 The Carathéodory number of X is the smallest k such that conv X =
convk X.
Remark 1.3 So, when X ⊂ Rn, Carathéodory’s theorem [
7
] is equivalent to the
equality conv X = convn+1 X. We will give an alternative definition for convk X in Sect. 4
as the kfold join of X.
Definition 1.4 A compactum X ⊂ Rn is kconvex if every linear image of X to Rk
is convex.
We give some examples of kconvex sets. What is needed in Fenchel’s theorem is
1convexity, and every connected set is 1convex. The kskeleton of a convex
polytope is kconvex (though for such kconvex sets, most results of this paper are trivial).
In [
6
] (see also [5, Chap. II, Sect. 14]) it is shown that the image of the sphere under
the Veronese map v2 : Sn−1 → Rn(n+1)/2 (with all degree 2 monomials as
coordinates) is 2convex.
In [12, Corollary 1] the following remarkable result is proved:
Theorem 1.5 (Hanner–Rådström, 1951) If X is a union of at most n compacta
X1, . . . , Xn in Rn and each Xi is 1convex, then convn X = conv X.
It is also known [
5, 15
] that a convex curve in Rn (that is, a curve with no n + 1
points in a single affine hyperplane) has Carathéodory number at most n +22 . It
would be interesting to obtain some nontrivial bounds for the Carathéodory
number of the orbit Gx of a point x in a representation V of a compact Lie group G in
terms of dim V and dim G (or the rank of G). The latter question is mentioned in [18,
Question 3] and would be useful in results like those in [17].
In Sects. 2 and 3 of this paper we show that the Carathéodory number is at most
k + 1 for (n − k)convex sets. In Sect. 4 we prove the corresponding analogue of the
colorful Carathéodory theorem, and in Sect. 5 we develop another topological
approach to colorful Carathéodorytype results, which may be of independent interest.
In Sect. 6 we give a related Tverbergtype result.
We note that the content of Sects. 2–4 may be trivially generalized to positive
hulls in place of convex hulls, as was done, for example, in [
2
]. But we do not state
the corresponding results explicitly to simplify the exposition.
2 The Carathéodory Number and kConvexity
We are going to give a natural generalization of the reasoning in [
12
]:
Theorem 2.1 Suppose that X1, . . . , Xn−k are compacta in Rn and p does not belong
to convk+1 Xi for any i. Then there exists an affine kplane L p that has empty
intersection with any Xi .
Remark 2.2 If we replace convk+1 Xi by the honest convex hull conv Xi , then the
result is simply deduced by induction from the Hahn–Banach theorem.
Remark 2.3 In [
16
] a somewhat related result was proved: For a compactum X ⊂ Rn
and a point p ∈/ X, there exists an affine kplane L (for a prescribed k < n) such that
the intersection L ∩ K is not acyclic modulo 2. Here acyclic means having the Cˇ ech
cohomology of a point.
The proof of Theorem 2.1 is given in Sect. 3. Now we deduce the following
generalization of Fenchel’s theorem [
11
] (stated in the second paragraph of the
introduction):
Corollary 2.4 If a compactum X ⊂ Rn is (n − k)convex, then convk+1 X = conv X.
Proof Assume the contrary and let p ∈ conv X \ convk+1 X. Applying Theorem 2.1
to the family X, . . . , X, we find a kdimensional L p disjoint from X. Now project
n−k
X along L with π : Rn → Rn−k . Since X is (n − k)convex, π(L) must be separated
from π(X) by a hyperplane. Hence, L is separated from X by a hyperplane, and
therefore p cannot be in conv X.
Remark 2.5 In the above Corollary 2.4 and its proof we could consider n − k
different (n − k)convex compacta X1, . . . , Xn−k and by the same reasoning obtain the
following conclusion:
n−k
i=1
convk+1 Xi =
conv Xi .
n−k
i=1
But this result trivially follows from Corollary 2.4 by taking the union.
Remark 2.6 For the image v2(Sn−1) of the Veronese map, the Carathéodory constant
is roughly of order n, see [5, Chap. II, Sect. 14, Theorem 14.3]. Hence, Corollary 2.4
is not optimal for this set.
3 Proof of Theorem 2.1
Let us replace Xi by a smooth nonnegative function ρi such that ρi > 0 on Xi and
ρi = 0 outside some εneighborhood of Xi . Let p be the origin.
Assume the contrary: For any kdimensional linear subspace L ⊂ Rn some
intersection L ∩ Xi is nonempty. The space of all possible L is the Grassmann
manifold Gkn. Denote by Di the open subset of Gkn consisting of L ∈ Gkn such that
L ρi > 0. Note that 0 cannot lie in the convex hull conv(L ∩ Xi ) because in this case
by the ordinary Carathéodory theorem, 0 would be in convk+1(L ∩ Xi ) ⊆ convk+1 Xi ,
contradicting the hypothesis. Hence (if we choose small enough ε > 0), the
“momentum” integral
mi (L) =
L
ρi x dx
never coincides with 0 over Di . Obviously, mi (L) is a continuous section of the
canonical vector bundle γ : E(γ ) → Gkn, which is nonzero over Di . Now we apply
the following:
Lemma 3.1 Any n − k sections of γ : E(γ ) → Gkn have a common zero because of
the nonzero Euler class e(γ )n−k .
This lemma is a folklore fact, see, for example, [
9, 22
]. Applying this lemma to
the sections mi , we obtain that the sets Di do not cover the entire Gkn. Hence, some
L ∈ Gkn has an empty intersection with every Xi .
Remark 3.2 In the proof of Theorem 1.5 in [12, the proof of Theorem 3] Hanner and
Rådström find a maximum of certain volume function over a convex subset of the
sphere Sn−1. This may also be done by an application of the Brouwer fixedpoint
theorem similar to the above proof, thus exhibiting the topological nature of that
result.
4 The Colorful Carathéodory Number
Let us introduce some notation and restate the colorful Carathéodory theorem [
2
].
Definition 4.1 Denote by A ∗ B the geometric join of two sets A, B ⊆ Rn, which is
t a + (1 − t )b : a ∈ A, b ∈ B, and t ∈ [
0, 1
] .
This is actually the alternative definition of convk X as X ∗ · · · ∗ X.
k
Theorem 4.2 (Bárány, 1982) If X1, . . . , Xn+1 ⊂ Rn are compacta and 0 ∈ conv Xi
for every i, then 0 ∈ X1 ∗ X2 ∗ · · · ∗ Xn+1.
It is possible to reduce the Carathéodory number n + 1 assuming the (n −
k)convexity of Xi , thus generalizing Corollary 2.4:
Theorem 4.3 Let 0 ≤ k ≤ n. If X1, . . . , Xk+1 ⊂ Rn are (n − k)convex compacta
and 0 ∈ conv Xi for every i, then 0 ∈ X1 ∗ X2 ∗ · · · ∗ Xk+1.
Proof We use the classical scheme [
2
] along with the degree reasoning used in [
1, 4,
8, 19
] in the proof of different generalizations of the colorful Carathéodory theorem.
First consider the case k = n − 1. In this case we have n sets and 1convexity. Let
x1, . . . , xn be the system of representatives of X1, . . . , Xn such that the distance from
S = conv{x1, . . . , xn} to the origin is minimal. If this distance is zero, then we are
done. Otherwise, assume that z ∈ S minimizes the distance.
Let z = t1x1 + · · · + tnxn, a convex combination of the xi s. If ti = 0, then we
observe that 0 ∈ conv Xi , and we can replace xi by another xi so that new simplex
S = conv{x1, . . . , xi−1, xi , xi+1, . . . , xn} is closer to the origin than S. So we may
assume that all the coefficients ti are positive and z is in the relative interior of S.
This also implies that S is (n − 1)dimensional, i.e., there is a unique hyperplane
containing S.
Consider the hyperplane h 0 parallel to S. Applying the definition of 1convexity
to the projection along h, we obtain that there exists a system of representatives yi ∈
Xi ∩ h. The set
f (B) = {x1, y1} ∗ {x2, y2} ∗ · · · ∗ {xn, yn}
is a piecewise linear image of the boundary of a crosspolytope, which we denote
by B. Note that for every facet F of B, the vertices of the simplex f (F ) form a system
of representatives for {X1, . . . , Xn}. In particular, S = f (F ) for some facet F of B.
The line through the origin and z intersects the simplex S = f (F ) transversally, and
so it must intersect some other f (F ) (where F = F is a facet of B) because of the
parity of the intersection index. The intersection ∩ f (F ) is on the segment [0, z]
and cannot coincide with z. Therefore, f (F ) is closer to the origin than S. This is a
contradiction with the choice of S. Thus, the case k = n − 1 is done.
The case k = 0 of this theorem is trivial by definition, and the case k = n
corresponds to the colorful Carathéodory theorem. Now let 0 < k < n − 1.
Consider again a system of representatives x1, . . . , xk+1 minimizing the distance
dist(0, conv{x1, . . . , xk+1}). Put S = conv{x1, . . . , xk+1}. As above, the closest to
the origin point z ∈ S must lie in the relative interior of S if z = 0.
Let L ⊂ Rn be the kdimensional linear subspace parallel to S. As in the first
proof, using (n − k)convexity, we select yi ∈ L ∩ Xi . Then we map naturally the
boundary B of a (k + 1)dimensional crosspolytope to the geometric join
f (B) = {x1, y1} ∗ {x2, y2} ∗ · · · ∗ {xk+1, yk+1}.
Note that f (B) is contained in the (k + 1)dimensional linear span of S and L, so by
the parity argument as above the image under f of some face of B must be closer to
the origin than S.
Remark 4.4 In this proof in the case k < n − 1 we can choose some (k +
1)dimensional subspace M ⊂ Rn and a system of representatives {x1, . . . , xk+1} for
M ∩ X1, . . . , M ∩ Xk+1. Then we can make the steps reducing dist(0, conv{x1, . . . ,
xk+1}) so that the system of representatives always remains in M .
5 A Topological Approach to Theorem 4.3
Theorem 4.3 can also be deduced from the following lemma:
Lemma 5.1 Let ξ : E(ξ ) → X be a kdimensional vector bundle over a compact
metric space X. Let Y1, . . . , Yk+1 be closed subspaces of E(ξ ) such that for every i, the
projection ξ Yi : Yi → X is surjective. If e(ξ ) = 0, then for some fiber V = ξ −1(x),
the geometric join
(Y1 ∩ V ) ∗ · · · ∗ (Yk+1 ∩ V )
contains 0 ∈ V .
Remark 5.2 The Euler class here may be considered in integral cohomology or in the
cohomology mod 2. The proof passes in both cases, so we omit the coefficients from
the notation.
Reduction of Theorem 4.3 to Lemma 5.1 for k < n Take a linear subspace M ⊆ Rn
of dimension k + 1. For every kdimensional linear subspace L ⊂ M , all the
intersections L ∩ Xi are nonempty. All such L constitute the canonical bundle γ over
k+1 = RP k with nonzero Euler class by Lemma 3.1. For any fixed i, the union of
Gk
sets L ∩ Xi constitutes a closed subset of E(γ ) that we denote by Yi . By Lemma 5.1,
for some L, the join
(Y1 ∩ L) ∗ · · · ∗ (Yk+1 ∩ L) = (X1 ∩ L) ∗ · · · ∗ (Xk+1 ∩ L)
must contain the origin.
Now we prove Lemma 5.1 The proof has much in common with the results of [
16
].
The main idea is that fiberwise acyclic (up to some dimension) subsets of the total
space of a vector bundle behave like sections of that vector bundle.
Let Y = Y1 ∗X ∗ · · · ∗X Yk+1 be the abstract fiberwise join over X, that is, the set
of all formal convex combinations
t1y1 + t2y2 + · · · + tk+1yk+1,
ξ(y1) = · · · = ξ(yk+1).
where ti are nonnegative reals with unit sum, and yi ∈ Yi are points such that
Denote the natural projection η : Y → X. Any formal convex combination y ∈ Y
defines a corresponding “geometric” convex combination f (y) in the fiber ξ −1(η(y))
depending continuously on y. It is easy to check that f (y) can be considered as a
section of the pullback vector bundle η∗(ξ ) over Y .
For any point x ∈ X, its preimage under η is a join of (k + 1) nonempty sets
Y1 ∩ ξ −1(x) ∗ · · · ∗ Yk+1 ∩ ξ −1(x) ,
and therefore η−1(x) is (k − 1)connected. Hence, the Leray spectral sequence for
the Cˇ ech cohomology H ∗(Y ) with E∗,∗ = H ∗(X; H∗(η−1(x))) (the coefficient sheaf
2
is the direct image of the homology of the total space) has empty rows number
1, . . . , k − 1, and its differentials cannot kill the image of e(ξ ) in Erk,0. Hence,
η∗(e(ξ )) = e(η∗(ξ )) remains nonzero over Y , and by the standard property of the
Euler class, for some y ∈ Y , the section f (y) must be zero.
Remark 5.3 In this proof we essentially use the inequality k < n. So the colorful
Carathéodory theorem is not a consequence of Lemma 5.1, at least in our present
state of knowledge.
The subsets Yi in Lemma 5.1 can be considered as setvalued sections. The same
technique proves the following:
Theorem 5.4 Let B be an ndimensional ball, and fi : B → 2B \ ∅ for i = 1, . . . ,
n + 1 be setvalued maps with closed graphs (in B × B). Then for some x ∈ B, we
have the inclusion
x ∈ f1(x) ∗ · · · ∗ fn+1(x).
Proof We may assume that all sets fi (x) are in the interior of B, because the general
case is reduced to this one by composing fi with a homothety with scale 1 − ε and
going to the limit as ε → +0.
It is known [
14
] that for a singlevalued map f : B → int B (considered as a
section of the trivial bundle B × Rn → B), a fixed point (x = f (x)) is guaranteed by
the relative Euler class e(f (x) − x) ∈ H n(B, ∂B). Then the proof proceeds as in
Lemma 5.1 by lifting e(f (x) − x) to the abstract fiberwise join of graphs of fi over
the pair (B, ∂B) and using the properties of the relative Euler class of a section.
Corollary 5.5 Suppose that X1, . . . , Xn+1 are compacta in Rn and ρ is a continuous
metric on Rn. For any x ∈ Rn, denote by fi (x) the set of farthest from x points in Xi
(in the metric ρ). Then, for some x ∈ Rn, we have
x ∈ f1(x) ∗ · · · ∗ fn+1(x).
Remark 5.6 If we denote by fi (x) the closets points in Xi , then this assertion
becomes almost trivial without using any topology.
6 The Carathéodory Number and the Tverberg Property
Tverberg’s classical theorem [
20
] is the following:
Theorem 6.1 (Tverberg, 1966) Every set of (n + 1)(r − 1) + 1 points in Rn can be
partitioned into r parts X1, . . . , Xr so that the convex hulls conv Xi have a common
point.
From the general position considerations it is clear that the number (n + 1) ×
(r − 1) + 1 cannot be decreased. But we are going to decrease it after replacing
a finite point set by a family of convex compacta. Let us define the Carathéodory
number for such families.
Definition 6.2 Let F be a family of convex compacta in Rn. The Carathéodory
number of F is the least κ such that for any subfamily G ⊆ F ,
conv
G =
conv
We denote the Carathéodory number of F by κ(F ).
Again, from the Carathéodory theorem [
7
] it follows that κ(F ) ≤ n + 1. Another
observation is that Corollary 2.4 guarantees that κ(F ) ≤ k + 1 if the union of every
subfamily G ⊆ F is (n − k)convex.
Now we state the analogue of Tverberg’s theorem.
Theorem 6.3 Suppose that F is a family of convex compacta in Rn, r is a positive
integer, and
Then F can be partitioned into r subfamilies F1, . . . , Fr so that
r
i=1
F  ≥ rκ(F ) + 1.
conv
Fi = ∅.
Remark 6.4 Note the following: If κ(F ) = n + 1, then taking a system of
representatives for F and applying the Tverberg theorem, we obtain a weaker condition:
F  ≥ (r − 1)(n + 1) + 1.
Remark 6.5 This theorem originated in discussions with Andreas Holmsen, who
established the same result in the special case n = 2, κ(F ) = 2, and with F  ≥ 2r (not
2r + 1), see [
13
]. Together with the previous remark, this shows that the condition on
F  may be not tight, though we have no idea how to improve it in the general case.
Proof of Theorem 6.3 We again use a minimization argument, combined with
Sarkaria’s trick [
21
] in the more convenient tensor form, which is from [
3
].
Let F  = m, κ = κ(F ), and
F = {C1, C2, . . . , Cm}.
Put the space Rn to A = Rn+1 as a hyperplane given by the equation xn+1 = 1.
Consider a set S of vertices of a regular (r − 1)simplex in some (r − 1)dimensional
space V and assume that S is centered at the origin.
Now define the subsets of V ⊗ A by
Xi = S ⊗ Ci
and consider a system of representatives (x1, x2, . . . , xm) for the family of sets G =
{X1, X2, . . . , Xm}. Such a system gives rise to a partition {Ps : s ∈ S} of {1, . . . , m}
in the following way. For s ∈ S, define
Ps = i ∈ {1, . . . , m} : xi = s ⊗ ci for some ci ∈ Ci .
Like in [1, Lemma 2], we observe that 0 ∈ conv{x1, . . . , xm} if and only if
s∈S conv{ci : i ∈ Ps } = ∅. Based on this, we choose a system of representatives
(x1, . . . , xm) of G so that the distance between 0 and conv{x1, x2, . . . , xm} is
minimal. If this distance is zero, then the required partition of F is given by the sets
{Ci ∈ F : i ∈ Ps }, s ∈ S.
Assume that the minimal distance is not zero. Then it is attained on some convex
combination
x0 = α1x1 + α2x2 + · · · + αmxm.
We claim that αi > 0 for all i ∈ {1, . . . , m}. Assume, for instance, that α1 = 0 and
x1 = s ⊗ c1 for some c1 ∈ C1 and s ∈ S. Now x1 can be replaced by t ⊗ c1 for
any t ∈ S as such a change does not influence x0. The distance minimality
condition implies that all the points t ⊗ c1 are separated from the origin by a hyperplane
in V ⊗ A, which is the support hyperplane for the ball, centered at the origin and
touching conv{x1, . . . , xm}. Obviously,
t∈S
t ⊗ ci = 0,
so the points t ⊗ ci , t ∈ S, are not separated from the origin. This contradiction
completes the proof of the claim.
The above convex combination representing x0 can be written as
i∈Ps αi = α(s). Thus,
Assume first that none of Ps is the empty set. Define c(s) = i∈Ps αi ci and α(s) =
i∈Ps αi > 0. Then c(s)/α(s) is a convex combination of elements ci ∈ Ci , i ∈ Ps .
Thus, c(s)/α(s) ∈ conv i∈Ps Ci . According to the definition of the Carathéodory
number, there is a subset Ps ⊂ Ps , of size at most κ , such that c(s)/α(s) ∈
conv i∈Ps Ci for every s ∈ S. This means that there are ci ∈ Ci for all i ∈ Ps such
that c(s)/α(s) ∈ conv{ci : i ∈ Ps }; in other words, c(s) = i∈Ps αi ci with positive αi
satisfying
x0 =
s ⊗
s∈S
i∈Ps
αi ci .
In this case the minimum distance is attained on the convex hull of no more that rκ
elements as each Ps  ≤ κ . But m > rκ , contradicting the claim.
Finally, we have to deal with the (easy) case where some Ps = ∅. The above
argument works, with no change at all, for the nonempty Ps , implying that x0 can be
written as a convex combination of at most (r − 1)κ elements. Again, m > (r − 1)κ ,
and the same contradiction finishes the proof.
Acknowledgements We thank Peter Landweber who has drawn our attention to those old results by
Fenchel, Hanner, and Rådström and Alexander Barvinok for discussions and examples of kconvexity.
The work of both authors was supported by ERC Advanced Research Grant No 267195 (DISCONV).
The first author acknowledges support from Hungarian National Research Grant No 78439. The second
author is supported by the Dynasty Foundation, the President’s of Russian Federation grant MD352.2012.1,
the Russian Foundation for Basic Research grants 100100096 and 100100139, the Federal Program
“Scientific and scientificpedagogical staff of innovative Russia” 2009–2013, and the Russian government
project 11.G34.31.0053.
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