# I have $n$-points that are on a unit sphere, how can I show using set builder notation that they are as far a way from each other as possible?

For example, if I had 2 points, they could be positioned in any way such that they are $pi$ radians away from each other in the x,y-direction, and for 3 points they would be $2\pi/3$ radians apart in the x,y-direction. I'm just having trouble showing this using set builder notation. Thank you.

• Is there a reason you want to express this in set-builder notation? – Brian Tung Aug 8 '18 at 21:04
• Well if there's any other way of expressing it that's fine, I'm just setting up a model and I'd like it to be defined mathematically as required by the criterion of the essay. – John Miller Aug 8 '18 at 21:06
• Maybe take a look at this – Jon Aug 8 '18 at 21:17

## 3 Answers

It's difficult to show this, but easy to express this!

You could characterize an extremal configuration ${\bf a}$ of $n$ points on $S^2$ by writing $${\bf a}\in {\rm argmax}_{\,{\bf x}\in (S^2)^n}\bigl(\min\nolimits_{1\leq i<j\leq n} \|x_i-x_j\|\bigr)\ .$$ I don't now whether this is more to the point than describing the idea in words.

• The point of the exercise was to write a paper in which everything needed to be written mathematically. I go to a French school and for some reason, my teacher is very strict with this, even though I find it a bit overkill. However, I believe this is the most concise formalisation of the idea. I assume it's just part of his teaching as it get's you to familiarise with complex set building notation. Thank you for taking your time in responding to my question. – John Miller Aug 9 '18 at 11:37

Reformatted after a comment by Arnaud Mortier.

Would something like the following do? $C = \{x_1, x_2, \ldots, x_n\}$ is a subset of $S^2$, the ordinary sphere, such that for all $C' = \{x'_1, x'_2, \ldots, x'_n\}$ a subset of $S^2$,

$$\min_{x, y \in C} d(x, y) \geq \min_{x', y' \in C'} d(x', y')$$

• Please don't write in a stream of consciousness manner. $S^2$ is not equal to a finite set. Even if everyone knows what you mean, this is giving very bad habits to people who are not mathematically mature yet. – Arnaud Mortier Aug 8 '18 at 21:28
• @ArnaudMortier: You know, I tend to prefer $C = {x_1, x_2, \ldots, x_n\} \subset S^2$, but for some reason I see that less and see things along the lines of what I have here more. Do you think the other is preferable? I'd rather use that. ETA: I don't know why my comment isn't MathJaxing. – Brian Tung Aug 8 '18 at 21:30
• I don't know either. Anyway, I believe it should be $\subset$ instead of $\in$. And yes, it is better imho. Students who write like this tend to be less able to organise their ideas correctly. – Arnaud Mortier Aug 8 '18 at 21:32
• @ArnaudMortier: Oops, yes, I meant $\subset$ in my comment, not $\in$—edited. – Brian Tung Aug 8 '18 at 21:33
• one of several similar problems: en.wikipedia.org/wiki/Tammes_problem – Will Jagy Aug 8 '18 at 21:59

Let $S$ be the set of points on the unit circle. Let $\mathbb P(A)$ be the Power Set (set of all subsets) of $A$. Let $\mathbb P_n(A)$ be defined as $$\{a\in\mathbb P(A)\; |\; |A| = n\}$$

Define $S'\subset\mathbb P_n(S^2)$ such that $$\{Q\in P_n(S)\;|\;\exists\epsilon\;\forall\;i,j\in Q\;i\neq j,\;d(i,j)=\epsilon\}$$

In layman's terms, $S'$ is the set of all sets of $n$ points on the circle such that the distance between any two of them is constant.

Define the function $F$ on elements of $\mathbb P(A)$ to be

$$F(s) = max\{d(i,j)\;|\;i,j\in s\}$$

So, the points you want will be those such that

$$\{s\in S'\;|\;F(s) = \sup\{F(s)\;|\;s\in S'\}\}$$