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.

  • $\begingroup$ In the plane containing two lines, there are usually two choices for angle. Do you want the larger choice or the smaller? If the former, use the same line n times for pi times (n choose 2) radians. If the latter, for n<=d go orthogonal and multiply the previous estimate by 1/2. Gerhard "Happy To Do Easy Cases" Paseman, 2014.07.09 $\endgroup$ – Gerhard Paseman Jul 9 '14 at 22:24
  • $\begingroup$ Also for d=2, n=3 rays, there are other optimal configurations, including all angles being 2pi/3. Gerhard "Is Not Always Right Thinking" Paseman, 2014.07.09 $\endgroup$ – Gerhard Paseman Jul 9 '14 at 22:28
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    $\begingroup$ The angle between two lines always does not exceed $\pi/2$, of course. Yes, there may be other optimal configurations, no uniqueness (if this is the answer). $\endgroup$ – Fedor Petrov Jul 9 '14 at 22:37
  • $\begingroup$ Are there local maxima in $d=3$ that are different from the above configuration? $\endgroup$ – Lev Borisov Jul 10 '14 at 0:52
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    $\begingroup$ I wonder if minimising the sum of the absolute values of the inner products of $n$ unit vectors will give the same extremal configurations. So you are minimising the sum of the cosines of the angles instead of maximising the sum of the angles. Just possibly it is more amenable to analysis. $\endgroup$ – Brendan McKay Jul 10 '14 at 3:04

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.

enter image description here


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