I have discovered the following results about the sums of squared distances between points on an $n$-sphere (and proved them). To the best of my knowledge (and my advisor's knowledge), these results are new. However, since this is classical geometry and the proofs are not hard, I wanted to ask if anyone knows of their existence in the literature.

I have searched the literature, and I do find authors considering similar questions (e.g., what distribution of points on an $n$-sphere maximizes the sum of distances between the points) but I do not see anyone considering the sum of *squared* distances.

I want to emphasize that I am *not* asking for proofs of these results. I have already proven them myself.

First, a bit of terminology:

- An
*$n$-sphere*is the set of all points equidistance from a fixed point in $\mathbb{R}^n$. - A set of points $\mathcal{V}$ is
*centrally symmetric*if it is closed under the antipodal map, i.e., for every $P \in \mathcal{V}$, the point $-P$ is in $\mathcal{V}$. - A set of points $\mathcal{V}$ is
*transitive*if for each $P, Q \in \mathcal{V}$, there exists a symmetry of $\mathcal{V}$ such that $P$ is mapped to $Q$.

**Result 1**. Let $\mathcal{V}$ be a set of $V$ points on a unit $n$-sphere, and let $\mathcal{C}$ be the multiset of the lengths of all the chords between them.
Then: $$\sum_{c \, \in \, \mathcal{C}}c^2= V^{2}(1-d^2)$$
where $d$ is the distance between the centroid of $\mathcal{V}$ and the center of the unit $n$-sphere.

This leads to a nice corollary: The centroid of $\mathcal{V}$ coincides with the center of the $n$-sphere *if and only if* $\sum_{c \, \in \, \mathcal{C}}c^2= V^{2}$.

**Result 2**. Let $\mathcal{V}$ be a transitive, centrally symmetric set of $V$ points on a unit $n$-sphere, and let $\mathcal{L}$ be the set of distinct lengths of all the chords between them.
Then: $$\sum_{l \, \in \, \mathcal{L}}l^2 = 2k+2$$
where $k$ is the cardinality of $\mathcal{L}$.

Question: Has anyone seen these results in the literature? If so, where? (I suppose reference to any similar results would be helpful as well...)

Note: My research was motivated by the following two facts found in the literature. My results subsume these (if one thinks of the vertices of a regular polygon as a point configuration).

- For a regular polygon inscribed in a unit circle, $\sum_{c \, \in \, \mathcal{C}}c^2= V^{2}$.
- For a regular polygon inscribed in a unit circle, $\sum_{l \, \in \, \mathcal{L}}l^2 \in \mathbb{Z}$.

Fact 1 is in an unpublished paper by S. Mustonen at https://www.survo.fi/papers/Roots2013.pdf. Fact 2 is in a popular math book by J. Kappraff: "Beyond Measure: A Guided Tour through Nature, Myth, and Number." (World Scientific Publishing, River Edge, NJ, 2002). A related question was stated as a 1923 MAA Monthly problem (Morley, F. V., Harding, A. M.: 2925. Am. Math. Mon. 30(1), 44 (1923)).

transitive(but that's OK, since you never use it). $\endgroup$ – Gerry Myerson Mar 20 at 2:00distinctchord lengths should be much messier than that. Does anyone have any intuition as to why thisshould havecome out as nice as it did? $\endgroup$ – Nathan Reading Mar 22 at 13:26