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.
Erdős' circle covering problem · Goodman–Goodman 1945
Let the circles $C_1, \dots, C_n$ with radii $r_1, \dots, r_n$ lie in a plane and have the following property: no line of the plane divides the circles into two non-empty sets without touching or intersecting at least one circle. Then the circles $C_1, \dots, C_n$ can be covered by a circle of radius $r = \sum_{i = 1}^n r_i$.
Arizona State University
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