Topological Method for

Colored Bárány's Theorem


姜子麟 Zilin Jiang

Point selection problem

How to define "median" for point set on plane?

Is there always a point in many triangles?

The planar case

Given $n$ points on the plane in general position.

Boros–Füredi Theorem

There exists a point in $\ge$$\frac{2}{9}$${n\choose 3}-O(n^2)$ triangles.


Bukh–Matoušek–Nivasch Theorem

The constant $\frac{2}{9}$ is the best possible by stretched grid!

Higher dimensional case

Given $n$ points in $\mathbb{R}^d$ in general position.

A simplex is the convex hull of $d+1$ points.

Bárány's Theorem

There exists a point in $\ge c_d{n \choose d+1}$ simplices.

Bukh–Matoušek–Nivasch Theorem

We cannot hope for $c_d > \frac{(d+1)!}{(d+1)^d}$ by stretched grid.

Timeline

Bárány: a point is in $\ge$ $c_d$$n\choose d+1$ simplices.

1982 Bárány $\frac{1}{(d+1)^d}$
2003 Wagner $\frac{(d^2+1)}{(d+1)^{d+1}}$
2010 Gromov $\frac{2d}{(d+1)(d+1)!}$ $< \frac{(d+1)!}{(d+1)^d}$

Plan of talk

Introduce a colored variant
Karasev's proof with a twist
images from boltcity.com

A colored variant

Given $n$ red points, $n$ green points and $n$ blue points.

Colored Bárány's Theorem

There is a point in at least $pn^3$ colorful triangles.

Probabilistic equivalence

Given $n$ red points, $n$ green points and $n$ blue points.

Probabilistic Equivalence

A point in random colorful triangle with probability $\ge p$.

Given 3 (continuous) probability measures $m_0$, $m_1$ and $m_2$.

Continuous Equivalence

A point in random colorful triangle with probability $\ge$ $p$.

Timeline (cont.)

2010 Gromov $\frac{2}{9}$ $\to \frac{2d}{(d+1)(d+1)!}$ $m_i = m_j$
2012 Karasev $\frac{1}{6}$ $\to \frac{1}{(d+1)!}$
$\frac{2}{9}$ $\to \frac{2d}{(d+1)(d+1)!}$ $m_1$ $=$ $m_2$
2014 J. $\frac{2}{9}$ $\to \frac{2d}{(d+1)(d+1)!}$

Plan of talk

Introduce a colored variant
Karasev's proof with a twist
images from boltcity.com

Karasev's proof

Given 3 (continuous) probability measures $m_0$, $m_1$ and $m_2$.

Proof by contradiction. Assume for every point $v$,

$\operatorname{Pr}\big(v \text{ is in} $ $ \big) < p := 2/9.$

Plane (plus a point at infinity) and sphere are the same.

$\infty$ $\equiv$ $\infty$

Preparation

  • Choose a sufficiently fine triangulation of sphere.
  • Add "buttress" inside the sphere.

Auxiliary map

Gradually build a map from "sphere with buttress" to sphere:

$\overset{f}{\longrightarrow}$

such that:

  1. it is identity when restricted to the sphere;
  2. it is economical: e.g., image of each "spike" is small.

Build economical map gradually

Step 1: Vertex $O$ (sphere center) $\mapsto \infty$ (the north pole).

Step 2: Want $\overline{PO}\mapsto \overline{P\infty}$; need to find line segment $\overline{P\infty}$.
Claim $\operatorname{Pr}\big(\overline{P\infty} \text{ intersects }$ $\big)$ $< p$ for some $\overline{P\infty}$.
Obs is in $\Longleftrightarrow$ intersects .

$\operatorname{Pr}\big($ intersects $\big)=\operatorname{Pr}\big($ is in $\big) < p$

$\implies \operatorname{Pr}\big($ intersects $\;\big\vert\; $$\big) < p$ for some

Step 3: Want to define $f$ on $OPQ$. Knew $f$ on $OP$, $OQ$, $PQ$.
$\overset{f}{\longrightarrow}$ $\overset{\mathrm{mod2}}{=}$

Set $a := m_1(A)$, $b := m_2(A)$, $x := \overline{m}(A) = (a + b)/2$.

$2x(1-x)\le $ $a(1-b)+b(1-a) \le$ $\operatorname{Pr}\big($ int. $\big)$ $< 2p$

$\implies x(1-x) < p = 2/9$. WLOG assume $\overline{m}(A) = x < 1/3$.

Degree argument

We have found a (continous) map $f$ such that

$\overline{m}\big(f(OPQ)\mathrm{mod2}\big)<\frac{1}{3}$ $\implies \overline{m}\big(f(\text{spike})\mathrm{mod2}\big) < 1$.

Def Degree of map $f$ is $\left|f^{-1}(\text{generic point})\right|$.

Note there are plenty of points not in $f(\text{spike})\mathrm{mod2}$.

Obs $\sum$deg of $f$ on each spike is even.
Cor $\text{deg on sphere} + 2\sum \text{deg on }OPQ$ is even.
! Degree of identity map is even. Impossible!