### Extremal Problems in Discrete Geometry

Arizona State University
July 23, 2021

Extremal properties of simple geometric objects

What's the minimum number of distinct distances between $n$ points on a plane?

Erdős 1946 · Moser 1952 · Chung 1984
Chung, Szemerédi & Trotter 1992 · Székely 1993
Solymosi & Tóth 2001 · Tardos 2003 · Katz & Tardos 2004
Guth & Katz 2015

Geometry of ruled surfaces, polynomial method.

Given $n$ points not all collinear in $\mathbb{R}^2$,
what's the minimum number of ordinary lines ?

Sylvester 1893 · Gallai 1944
Dirac & Motzkin 1951 · Kelly & Moser 1958
Csima & Sawyer 1993 · Green & Tao 2012

Menelaus's theorem, Euler formula
Cayley–Bacharach theorem, sum-product estimate

What's the densest way to arrange
non-overlapping spheres in $\mathbb{R}^n$?

Kepler 1611 · Gauss 1831 · Thue 1890
Fejes Tóth 1940 · Hales 1998
Cohn, Kumar, Miller, Radchenko, Viazovska 2016

Poisson summation formula, modular forms, automated proof checking

How high do $n$ satellites need to fly?

Each satellite sees a zone of the sphere

What's the minimum width of $n$ equal zones
covering the unit sphere?

Is it $\pi / 3$ for $n = 3$?

Distribute $n$ great circles to minimize the greatest distance between a point and the nearest great circle.

What's the minimum width of $n$ equal zones
covering the unit sphere?

Fejes Tóth's zone conjecture

The width of $n$ equal zones covering the
unit sphere is at least $\pi / n$.

Research Problems: Exploring a Planet, 1973

Rosta 1972: 3 equal zones
Linhart 1974: 4 equal zones
Fodor, Vígh and Zarnócz 2016: a lower bound
for example, $\pi/6.83$ for 5 equal zones

Fejes Tóth's zone conjecture 1973 · J.–Polyanskii 2017

The total width of $n$ equal zones covering
the unit sphere is at least $\pi$ in any dimension; Characterization of equality cases.

A plank in $\mathbb{R}^d$ is

Tarski's plank problem

What's the minimum total width of planks covering a given convex body?

What's the minimum total width of planks covering a unit disk?

Plank of width $w$

Planks cover disk

$\sum w_i \ge 2$

Archimedes: Arch of area $\pi w$

Arches cover hemisphere

$\sum \pi w_i \ge 2\pi$

What's the minimum total width of planks
covering a given convex body $C$?

Tarski's plank conjecture 1932 · Bang 1951

If convex body $C$ is covered by planks, then their total width is at least the width of $C$.

A direction of a plank is

Bang's lemma

Can choose directions such that $v_1 + \dots + v_n$ is not covered

Fejes Tóth's zone conjecture 1973 · J.–Polyanskii 2017

The total width of $n$ zones covering the unit sphere is at least $\pi$.

Each zone is a plank $\cap$ the unit sphere
and it has two directions

Bang's lemma: $v := v_1 + \dots + v_n$ is not covered
If $\lVert v \rVert \le 1$, then $\hat{v}$ is not covered.

Otherwise, $v$ is large in magnitude
We merge some zones

Erdős' circle covering problem · Goodman–Goodman 1945

A non-separable family of balls of radii $r_1, \dots, r_n$ can be
covered by a ball of radius $r_1 + \dots + r_n$

Bang's plank conjecture 1951

Total relative width of planks covering
a convex body is at least $1$.

Bezdek's annulus conjecture 2003

Total width of planks covering an annulus with a small hole at center is at least the diameter.

Ball 1992 using Bang's lemma

A lower bound on sphere packing density.

Equiangular lines in $\mathbb{R}^n$ are lines through origin pairwise separated by the same angle

What's the maximum number of
equiangular lines in $\mathbb{R}^n$?

 $n$ 2 3-4 5 6 7-14 ... 23-41 42 43 max 3 6 10 16 28 ... 276 276-288 344

What's the maximum number of equiangular lines in $\mathbb{R}^n$?

Gerzon 1973

At most $\frac{1}{2}n(n+1)$

de Caen 2000

At least $cn^2$

Angles $\to$ 90° as $n\to\infty$

What happens if the angles are held fixed?

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

 Lemmens, Seidel 1973 $E_{1/3}(n) = 2(n-1)$ for $n \ge 15$ Neumann 1973 $E_{\alpha}(n) \le 2n$, unless $1/\alpha$ is odd Neumaier 1989 $E_{1/5}(n) = \lfloor \frac{3}{2}(n-1) \rfloor$ for $n \ge n_0$ Bukh 2016 $E_{\alpha}(n) \le c_\alpha n$ Balla, Dräxler, Sudakov, Keevash 2018 $E_{\alpha}(n) \le 1.93n$for $n \ge n_0(\alpha)$ if $\alpha \neq 1/3$

Bukh's conjecture on equiangular lines with fixed angle

$E_{1/(2k-1)}(n) \approx \frac{kn}{k-1}$. $E_{1/7}(n) \approx \frac{4}{3}n$.

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

Conjecture J.–Polyanskii

$E_{\alpha}(n) \approx \frac{kn}{k-1}$, where $k = k(\lambda)$, $\lambda = \frac{1-\alpha}{2\alpha}$.

Spectral radius order $k(\lambda) :=$ smallest $k$ such that
$\exists$ $k$-vertex graph $G$ whose adjacency matrix has spectral radius $= \lambda$

 $\alpha$ $\lambda$ $G$ $k$ $E_\alpha(n)$ $\tfrac{1}{3}$ $1$ $2$ $2n$ $\tfrac{1}{5}$ $2$ $3$ $\tfrac{3n}{2}$ $\frac{1}{7}$ $3$ $4$ $\tfrac{4n}{3}$

J.–Polyanskii 2018

True for all $\lambda \le \sqrt{2 + \sqrt{5}}$.

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

me
Jonathan Tidor
Yufei Zhao
Yuan Yao
Shengtong Zhang

J., Tidor, Yao, Zhang, Zhao 2019

$E_\alpha(n) = \lfloor \frac{k}{k-1}(n-1) \rfloor$ for $n \ge n_0(\alpha)$ if $k(\lambda) < \infty$;
$E_\alpha(n) = n+o(n)$ otherwise.

J., Tidor, Yao, Zhang, Zhao 2019

Multiplicity of second largest eigenvalue of an $n$-vertex connected graph of bounded degree is at most $Cn/\log\log n$.

Cayley graph of $\mathrm{PSL}(2,p)$ has
second eigenvalue multiplicity at least $\sqrt[3]{n}$

What is the maximum second eigenvalue multiplicity of
an $n$-vertex connected graph of bounded degree?

What is the maximum second eigenvalue multiplicity of
an $n$-vertex Cayley graph of bounded degree?

Gromov, Colding–Minicozzi, Kleiner
Lee–Makarychev: Constant for abelian groups

What is the maximum number of unit vectors in $\mathbb{R}^n$ such that pairwise inner products are either $\alpha$ or $\beta$?

Zauner's conjecture 1999

Maximum number of equiangular lines in $\mathbb{C}^n$ is $n^2$.