High-dimensional phenomena in discrete analysis

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4. Discrete geometry

Equi-isoclinic subspaces

Given $k$ -dimensional linear subspaces $U,V\leq \mathbb{R}^d$ , we say that $U,V$ are equi-isoclinic with angle $\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\alpha$ .

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Conjecture 4.1.
[Yufei Zhao] Let $N_\alpha^k(d)$ be the maximum number of $k$ -dimensional subspaces of $\mathbb{R}^d$ , every pair of which is equi-isoclinic with angle $\alpha$ . Then

(1) $N_{\alpha}^k(d)=O_{\alpha,k}(d)$ .

(2) $N_{\alpha}^k(d)=O_{k}(d)+o_{\alpha,k}(d)$ .
The case $k=1$ gives equiangular lines, which is known, using Ramsey Theory.

$k=2$ generalizes "complex equiangular lines". Conjecture 1 is known for complex equiangular lines.
Distinct distances and related problems

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Conjecture 4.2.
[Cosmin Pohoata] (1) Let $S\subset\mathbb{R}^2$ with no three points collinear. Then $S$ determines at least $|S|/2$ distinct distances.

(2) Given $n$ points in $\mathbb{R}^2$ with no three collinear. Then one can remove $O(1)$ -many points so that the number of isosceles triangles is at most $1.99\binom{n}{2}$ .

(3) Given a set $S$ of $n$ points in $\mathbb{R}^2$ with no three collinear. Then $S$ determines $\Omega(n^2)$ perpendicular bisectors containing at most 1 point of $S$ each, unless $S$ is the vertices of a regular polygon together with its center.
(1) is known with $|S|/3$ distinct distances, by double counting isosceles triangles.
Kupitz conjecture in three dimensions

(Kupitz conjecture) Given $n$ points in $\mathbb{R}^2$ , is there always a line through at least two points, such that the number of points on each side differ by $O(1)$ ?

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Problem 4.3.
[Jeck Lim] Given $n$ points in $\mathbb{R}^3$ , is there always a plane through at least two points, such that the number of points on each side differ by $O(1)$ ?
For Kupitz conjecture, if the optimal bound is $C$ , then it is known that $2\leq C\leq O(\log\log n)$ . For pseudoline arrangements, $C=\Theta(\log\log n)$ .

For 3D, the same upper bound $O(\log\log n)$ follows by projection.

Cite this as: AimPL: High-dimensional phenomena in discrete analysis, available at http://aimpl.org/highdimdiscrete.

High-dimensional phenomena in discrete analysis

4. Discrete geometry

Equi-isoclinic subspaces

Conjecture 4.1.

Distinct distances and related problems

Conjecture 4.2.

Kupitz conjecture in three dimensions

Problem 4.3.