### Colored Bárány's theorem

Massachusetts Institute of Technology
July 22, 2019

### Point selection problem

Median for plane? 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.

### 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 $c_d > \frac{(d+1)!}{(d+1)^d}$.

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

### Colored variant

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

Colored Bárány's Theorem

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

Probabilistic Equivalence

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

There exists a point in a random colorful triangle with probability $\ge p$.

Continuous Equivalence

Given probability measures $m_0$, $m_1$and $m_2$.

There exists point in a random colorful triangle with probability $\ge$ $p$.

 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$$= m_2 2014 J. \frac{2}{9} \to \frac{2d}{(d+1)(d+1)!} ### Plan of talk Introduce a colored variant Karasev's proof with a twist ### Karasev's proof Given probability measures m_0, m_1and m_2. Proof by contradiction. Assume for every point v, \operatorname{Pr}\big(v \text{ is in} \big) < p := 2/9. Plane (plus a point at infinity) and sphere are the same. \infty$$\equiv$$\infty ### Auxiliary map Gradually build a map such that: \overset{f}{\longrightarrow} 1. it is identity when restricted to the sphere; 2. it is economical: image of each "spike" is "small". ### Build economical map gradually Step 1: Vertex O (sphere center) \mapsto \infty (the north pole). Step 2: Want \overline{PO}\mapsto \overline{P\infty}. Claim \operatorname{Pr}\big(\overline{P\infty} \text{ intersects } \big) < p for some \overline{P\infty}. Observation P is in \Longleftrightarrow intersects . \operatorname{Pr}\big( intersects \big)=\operatorname{Pr}\big(is in \big) < p \implies \operatorname{Pr}\big(intersects \;\big\vert\;$$\big) < p$ for some

Step 3: Want to define $f$ on $OPQ$.

$\overset{f}{\longrightarrow}$$\overset{\mathrm{mod2}}{=} Set a = \overline{m}(A) = (m_1(A) + m_2(A))/2. 2a(1-a)\le \operatorname{Pr}\big( intersects \big)$$< 2p$
$\implies a(1-a) < p = 2/9$. WLOG assume $a < 1/3$.

### Degree argument

Found a 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$.

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

$\sum$deg of $f$ on each spike is even.

$\text{deg on sphere} + 2\sum \text{deg on }OPQ$ is even.

Degree of identity map is even.

### Open problems

$c_3 = ?$

$0.07480 \le c_3 \le 0.09375$

Lower bound by Král', Mach and Sereni

Selection problem in convexity space

In a convexity space with Radon number $3$, does there exist a point in $\frac{1}{2}\binom{n}{2}$ segments?

Massachusetts Institute of Technology
zilinj@mit.edu