A Slight Improvement to the Colored B\'ar\'any's Theorem

Suppose $d+1$ absolutely continuous probability measures $m_0, \ldots, m_d$ on $\mathbb{R}^d$ are given. In this paper, we prove that there exists a point of $\mathbb{R}^d$ that belongs to the convex hull of $d+1$ points $v_0, \ldots, v_d$ with probability at least $\frac{2d}{(d+1)!(d+1)}$, where each point $v_i$ is sampled independently according to probability measure $m_i$.


Introduction
Let P ⊂ R d be a set of n points. Every d + 1 of them span a simplex, for a total of n d+1 simplices. The point selection problem asks for a point contained in as many simplices as possible. Boros and Füredi [BF84] showed for d = 2 that there always exists a point in R 2 contained in at least 2 9 n 3 − O(n 2 ) simplices. A short and clever proof of this result was given by Bukh [Buk06]. Bárány [Bár82] generalized this result to higher dimensions: Theorem 1 (Bárány [Bár82]). There exists a point in R d that is contained in at least c d This general result, the Bárány's theorem, is also known as the first selection lemma. We will henceforth denote by c d the largest possible constant for which the Bárány's theorem holds true. Bukh, Matoušek and Nivasch [BMN10] used a specific construction called the stretched grid to prove that the constant c 2 = 2 9 in the planar case found by Boros and Füredi [BF84] is the best possible. In fact, they proved that c d d!
(d+1) d . On the other hand, Bárány's proof in [Bár82] implies that c d (d + 1) −d , and Wagner [Wag03] improved it to c d d 2 +1 (d+1) d+1 . Gromov [Gro10] further improved the lower bound on c d by topological means. His method gives c d 2d (d+1)(d+1)! . Matoušek and Wagner [MW11] provided an exposition of the combinatorial component of Gromov's approach in a combinatorial language, while Karasev [Kar12] found a very elegant proof of Gromov's bound, which he described as a "decoded and refined" version of Gromov's proof.
The exact value of c d has been the subject of ongoing research and is unknown, except for the planar case. Basit, Mustafa, Ray and Raza [BMRR10] and successively Matoušek and Wagner [MW11] improved the Bárány's theorem in R 3 . Král', Mach and Sereni [KMS12] used flag algebras from extremal combinatorics and managed to further improve the lower bound on c 3 to more than 0.07480, whereas the best upper bound known is 0.09375. However, in this paper, we are concerned with a colored variant of the point selection problem. Let P 0 , . . . , P d be d + 1 disjoint finite sets in R d . A colorful simplex is the convex hull of d + 1 points each of which comes from a distinct P i . For the colored point selection problem, we are concerned with the point(s) contained in many colorful simplices. Karasev proved: By a standard argument which we will provide immediately, a result on the colored point selection problem follows: Corollary 3. If P 0 , . . . , P d each contains n points, then there exists a point that is contained in at least 1 (d+1)! · n d+1 colorful simplices.
Our result drops the additional assumption in theorem 2, hence improves corollary 3: Main Theorem. There is a point in R d that belongs to an m-simplex with probability p d  Proof of corollary 4 from the main theorem. Given d + 1 sets P 0 , . . . , P d in R d each of which contains n points. Let Ψ : R d → R be the bump function defined by Ψ(x 1 , . . . , It is a standard fact that Ψ and Ψ n are absolutely continuous probability measures supported on [−1, 1] d and [−1/n, 1/n] d respectively.
For each n ∈ N and 0 k d, define m k is an absolutely continuous probability measure supported on the Minkowski sum of P k and [−1/n, 1/n] d . Let m (n) be the family of d + 1 probability measures m Because no point in a certain neighborhood of infinity is contained in any m (n) -simplex, the set {p (n) : n ∈ N} is bounded, and consequently the set has a limit point p. Suppose p is contained in N colorful simplices. Let ǫ > 0 be the distance from p to all the colorful simplices that do not contain p. Choose n large enough such that 1/n ≪ ǫ and Readers who are familiar with Karasev's work [Kar12] would notice that our proof of the main theorem heavily relies on his arguments. The author is deeply in debt to him.

Proof of the Main Theorem
In this section, we provide the proof of the main theorem. The topological terms in the proof are standard, and can be found in [Mat03]. In addition to the notion of an m-simplex, in the proof, we will often refer to an (m k , . . . , m d )-face which means the convex hull of d − k + 1 points v k , . . . , v d with each point v i sampled independently according to probability measure m i . An m-simplex and an (m k , . . . , m d )-face are both set-valued random variables.
Proof of the main theorem. To obtain a contradiction, we suppose that for any point v in R d , the probability that v belongs to an m-simplex is less than p d := 2d (d+1)(d+1)! . Since this probability, as a function of point v, is continuous and uniformly tends to 0 as v goes to infinity, there is an ǫ > 0 such that v is contained in an m-simples with probability at most be the one-point compactification of the Euclidean space R d . Take δ = ǫ/d. Choose a finite triangulation 2 T of S d with one of the d-simplices containing ∞ such that for 0 < k d, any k-face of T intersects an (m k , . . . , m d )-face with probability less than δ and that the measure of 2 A triangulation T of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h : ||K|| → X. Since the finite triangulation of interest is an extension of the triangulation of a d-simplex X in R d and h is an identity map, we will freely use topological notions such as "a k-face (as a subset of S d )" instead of "the image of a k-face in K under h". With such abuse of language, we can avoid going back and forth between the simplicial complex and the topological space. any d-face of T under (m d−1 + m d ) /2 is less than δ. This can be done by taking a sufficiently fine triangulation of S 2 with one d-simplex having ∞ in its relative interior.
We use cone(·) as the cone functor 3 with apex O. A triangulation T of S d naturally extends to a triangulation cone(T ) of cone(S d ). We denote the k-skeleton 4 of T and cone(T ) by T k and cone(T ) k respectively.
We are going to define a continuous map f : cone(T ) d → S d . Put f (x) = x for all x ∈ S d = ||T || ⊂ cone(T ) d , and set f (O) = ∞. We proceed to define f on cone(σ) for all the k-faces σ of T inductively on dimension k of σ while we maintain the property that the image of the boundary of cone(σ) under f , that is f (∂cone(σ)), intersects an (m k , . . . , m d )-face with probability at most (k + 1)!(p d − ǫ + kδ). We say f is economical over a k-face σ of T d−1 if f and σ satisfy the above property. Unlike Karasev [Kar12], our inductive construction of f follows the same pattern until k = d − 2 instead of d − 1. The main innovation of this proof is a different construction for k = d − 1, which enables us to remove the additional assumption in theorem 2.
Suppose f is already defined on cone(T ) k and it is economical over k-faces of T . We are going to extend the domain of f to cone(T ) k+1 . Indeed, we only need to define f on cone(σ) for every k-face σ of T . (m k , . . . , m d )-face. Notice that the following statements are equivalent: • f (∂cone(σ)) intersects conv(v k , . . . , v d ); • for some v ∈ f (∂cone(σ)), the ray with initial point v in the direction # » v k v intersects conv(v k+1 , . . . , v d ).
Now, we define f on cone(σ). First, let g be the homeomorphism from cone(σ) onto the cone over ∂cone(σ) with apex c such that g is an identity on ∂cone(σ). This can be done because cone(σ) is homeomorphic to a (k + 1)-simplex ∆ and it is easy to find a homeomorphism from ∆ to cone(∂∆) that keeps ∂∆ fixed.   Define f on cone(σ) to be the composition of g and h: According to the commutative diagram above, f is well-defined on cone(σ) in the sense that it is compatible with its definition on cone(T ) k . We use the phrase "fill in the boundary of cone(σ) against the center v σ k " to represent the above process that extends the domain of f from ∂cone(σ) to cone(σ).
In other words, one ofm(A) andm(B) is less than 1−δ d+1 . We may assume thatm(B) < 1−δ d+1 . Fix a point c ∈ A. Again, we fill in the boundary of cone(σ) against the center c. For any generic point x ∈ A, the line segment [c, x] intersects with D an even number of times. For every v on ∂cone(σ), the ray with the initial point f (v) in the direction # » cf (v) covers x once if and only if the line segment [c, x] intersects with D at f (v). Because f (cone(σ)) is the union of such rays, the number of times that x is covered by f (cone(σ)) is exactly the number of intersections between [c, x] and D. This implies that x is not in f (cone(σ))mod2. Therefore f (cone(σ))mod2 is a subset of B ∪D almost surely. Noticing thatm(D) = 0, the extension of f has the desired propertym (f (cone(σ))mod2) < 1−δ d+1 . ⊂ τ ∪ f (cone(σ 0 ))mod2 ∪ . . . ∪ f (cone(σ d ))mod2.
Thereforem (f (∂cone(τ ))mod2) is less than δ + (d + 1) 1−δ d+1 = 1, and so the degree of f on ∂cone(τ ), denoted by deg (f, ∂cone(τ )), is even. Because where the first sum and the second sum are over all d-faces and all (d − 1)-faces of T respectively, we know that deg (f, T ) is even, which contradicts with the fact that f is identity on T .