$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Let $B\_d$ and $S\_{d-1}$ denote, respectively, the closed unit ball and the unit sphere in $\R^d$. Let us say that a finite subset $F$ of $B\_d$ is maximal if the sum of all pairwise Euclidean distances between the points in the set $F$ is the largest possible given $n:=\text{card}\,F$.


In a comment to a post now apparently deleted, Noam D. Elkies noted that, by convexity, (i) any maximal set must lie on $S\_{d-1}$ and (ii) such a set must be the set of all vertices of a regular polygon if $d=2$.


This suggests the following questions:


**Q1.** For $d=3$ and $n$ such that there exists a [regular polytope](https://en.wikipedia.org/wiki/Regular_polytope#Regular_complex_polytopes) with $n$ vertices, is it true that every maximal set must be the set of all vertices of a regular polytope?


**Q2.** For $d=3$ and any given natural $n$, is a maximal set of $n$ distinct points unique up to an orthogonal transformation?


**Q3.** A weaker version of Q2: For $d=3$ and any given natural $n$, are there only finitely many, up to an orthogonal transformation, maximal sets of $n$ distinct points?


**Q4.** Same questions for any $d\ge3$.


**Q5.** Same questions with the Euclidean distances $|x-y|$ between points $x$ and $y$ replaced by increasing functions $f(|x-y|)$ of the distances; e.g., here one may take $f(r)\equiv r^2$ or $f(r)\equiv -1/r^{d-2}$.


The answer to the questions about regular polytopes seem to be positive at least for $n\le d+1$, with the regular simplexes as the regular polytopes.




---


One of the correct and complete answers to any one of these questions will be accepted as an answer to this entire post.



Subsets of a ball/sphere with the largest sum of distances
Negative answers to some of those questions:


**Q1** Not always; for $n=8$ a square antiprism is better than a cube.
For example, in radius $\sqrt 3$, the cube with vertices $(\pm 1, \pm1, \pm1)$
gives $16(1 + 2\sqrt{2} + \sqrt{3}) = 88.9676+$ while changing
the $z=+1$ vertices to $(\pm\sqrt2, 0, 1)$ and $(0,\pm\sqrt2,1)$ gives
$16\bigl(\!\sqrt{8+\sqrt{8}} + \sqrt{8-\sqrt{8}}\bigr) = 89.0362+$. One can then
optimize the $z$-values further. [Similarly, in ions of the form
${\rm MF}\_8^{n-}$, with eight fluorine atoms around a large central M atom
(such as xenon [sic] or rhenium), the F's form a square antiprism, not a cube.]


**Q5** Again not always. Indeed for $\sum|x-y|^2$ we have
$$
\mathop{\sum\!\sum}\_{1 \leq i < j \leq n} 
 |x\_i - x\_j|^2 = n^2 - \Bigl| \sum\_i x\_i \Bigr|^2
$$
so the maximal configurations are precisely those whose center of mass
is the origin, and once $n$ is at all large there is a positive-dimensional
family of such configurations. There are also examples for other natural $f$
where the 24-cell (the $(d,n)=(4,24)$ regular polytope) is suboptimal; see



> 
> Henry Cohn, John H. Conway, Noam D. Elkies, and Abhinav Kumar: The $D\_4$ root system is not universally optimal, *Experimental Math.* **16** (2007), 313--320,
> [arXiv:math/0607447](https://arxiv.org/abs/math/0607447).
> 
> 
> 


