maximum sum of angles between $n$ lines

Question

Take $n$ lines in $\mathbb{R}^d$ (not necessary different, and all passing through the origin, though this is not important). What is maximal possible sum of angles between them for given $n$ and $d$? Conjecturally you should use only $d$ different mutually orthogonal directions, either $\lfloor n/d \rfloor$ or $\lceil n/d \rceil$ times each of them. I can not prove or disprove this even for $d=3$. Note that if you take $n$ rays instead of lines then the optimal configuration contains only two opposite directions, each $\lfloor n/2 \rfloor$ or $\lceil n/2 \rceil$ times.

Answer 1

This problem seems to be a question of Fejes Tóth posed here:

L. Fejes Tóth, ``Über eine Punktverteilung auf der Kugel", Acta Math. Acad. Sci. Hungar 10 (1959), 13-19 (in German).

Some recent work on this question in $\mathbb R^3$ has been done in

F. Fodor, V. Vígh, and T. Zarnócz, ``On the angle sum of lines", Arch. Math. 106 (2016), 91-100.

Amusingly, the above paper, in addition to the original conjecture by Fejes Tóth, references this mathoverflow post.

Let $Z=\{z_1, ..., z_N\}$ be $N$ points on the sphere $\mathbb S^{d-1} \subset \mathbb R^d$. Let $F(t) = \arccos |t|$, then the ``line" (acute) angle between two vectors $z_i$ and $z_j$ is $F(z_i \cdot z_j)$. We are trying to maximize the discrete energy $$ E_F (Z) = \frac{1}{N^2} \sum_{i,j =1}^N F(z_i \cdot z_j).$$ The conjecture then states that the maximum of this energy is equal to $\displaystyle{\frac{\pi}2 \cdot \frac{d-1}{d}}$ when $N$ is a multiple of $d$ (with the necessary correction for other cases).

More generally one can consider the energy integral $$ I_F (\mu ) = \int_{\mathbb S^d}\int_{\mathbb S^d} F(x \cdot y ) d\mu (x) d \mu (y) ,$$ where $\mu $ is a probability measure on the sphere. It is also reasonable to conjecture that $\max I_F(\mu ) = \displaystyle{\frac{\pi}2 \cdot \frac{d-1}{d}}$, i.e. the energy is maximized if $\mu$ is equally concentrated in vertices of an orthonormal basis.

In this terms, the paper of Fodor, Vígh, Zarnócz proves that in dimension $d=3$ for any point distribution $$ E_F (Z) \le \frac{3\pi}8, $$ (with a small correction for $N$ odd), while the conjecture in $d=3$ is $\frac{\pi}3$.

Ryan Matzke and I recently realized that there is a simple way to improve this bound. Here is the argument: one can easily check that for $-1\le t \le 1$ $$ F (t) = \arccos |t| \le \frac{\pi}{2} - \frac{7 \pi}{16} t^2 $$ (see the graph in the end). Therefore, $$ I_F (\mu) \le \frac{\pi}2 - \frac{7 \pi}{16} I_{t^2} (\mu) \le \frac{\pi}2 - \frac{7 \pi}{16 d} ,$$ where we have used the fact that $I_{t^2} (\mu ) \ge \frac1{d} $ (this is a well-known lower bound on the ``frame energy", it is essentially proved in fedja's reply to the aforementioned mathoverflow post: The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$).

Hence for $d=3$ we obtain $$ I_F (\mu ) \le \frac{17 \pi}{48},$$ which is better than $\frac{3\pi}{8}$, and is only $\frac{\pi}{48}$ away from the conjectured $\frac{\pi}{3}$.

Here is a picture that ``proves" the inequality above. There is a tiny bit of room for improvement, but his will definitely not yield the conjecture.