### Open problems in discrete geometry

Arizona State University
September 28, 2023

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 $\pi / n$.

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 $C$.

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.

Bezdek

Let $R$ be a region obtained from the unit square by removing an axis-parallel square of side 1/2 from its interior. Then the total width of planks covering $R$ is at least 1.

Reformulation of Bang's plank conjecture 1951

Let $C$ be a convex set inscribed in the unit cube $[0, 1]^d$, and let $I_1, I_2, \dots, I_d \subset [0, 1]$ be intervals. If for every $x \in C$ there exists $i$ such that $x_i ∈ I_i$, then $\sum_{i=1}^d \lvert I_i \rvert \ge 1$.

Aharoni, Holzman, Krivelevich, and Meshulam 2022

Let $C$ be a convex set inscribed in the unit cube $[0, 1]^d$, and let $f_1,\dots, f_d$ be non-negative measurable functions on $[0,1]$. If $\sum_{i=1}^d f(x_i) \ge 1$ holds for every $x \in C$, then $\sum_{i=1}^d \int f_i \ge 2/d$.

Zhao 2022

If a polynomial $p \in \mathbb{R}[x_1, \dots , x_d]$ of degree $n$ has a nonzero restriction to the unit sphere $S^{d-1} \subset \mathbb{R}^d$ and attains its maximum absolute value on $S^{d-1}$ at a point $x$ then $x$ is at angular distance at least $π/(2n)$ from the intersection of the zero set of $p$ with $S^{d-1}$.

Ortega-Moreno 2021

If a polynomial $p \in \mathbb{C}[x_1, \dots , x_d]$ of degree $n$ has a nonzero restriction to the unit sphere $S^{2d-1} \subset \mathbb{C}^d$ and attains its maximum absolute value on $S^{2d-1}$ at a point $x$ then $x$ is at angular distance at least $\arcsin(1/\sqrt{n})$ from the intersection of the zero set of $p$ with $S^{2d-1}$.

Glazyrin, A. Polyanskii, and R. Karasev 2022

If $p_1, \dots,p_n \in \mathbb{R}[x_1,\dots,x_d]$ are nonzero polynomials and $\delta_1, \dots ,\delta_n > 0$ are such that $\sum_{i=1}^{n}\delta_k \deg p_k \le 1$, then there exists $x \in B^d \subset \mathbb{R}^d$ such that for every $k \in \{1,\dots,n\}$, the point $x$ is at distance at least $\delta_k$ from the zero set of $p_k$.

Glazyrin, A. Polyanskii, and R. Karasev 2022

If $p_1, \dots,p_n \in \mathbb{R}[x_1,\dots,x_d]$ have nonzero restrictions to $S^{d-1} \subset \mathbb{R}^d$ and $\delta_1, \dots ,\delta_n > 0$ are such that $\sum_{i=1}^{n}\delta_k \deg p_k \le \pi/2$, then there exists $x \in S^{d-1} \subset \mathbb{R}^d$ such that for every $k \in \{1,\dots,n\}$, the point $x$ is at angular distance at least $\delta_k$ from the intersection of the zero set of $p_k$ with $S^{d-1}$.

Arizona State University
zilinj@asu.edu