Let $X_n$ denote the set of all collections $(v_{ij})$ of points on the sphere $S^2$, for $1 \leq i,j \leq n$, $i \neq j$, such that:

  1. $v_{ji} = -v_{ij}$,

  2. the origin $O$ in $\mathbb{R}^3$ is in the convex hull of $v_{ab}$, $v_{bc}$ and $v_{ca}$ for all $1 \leq a < b < c \leq n$.

  3. the origin $O$ in $\mathbb{R}^3$ is in the convex hull of $v_{ab}$, $v_{bc}$, $v_{cd}$ and $v_{da}$, for all $1 \leq a < b < c < d \leq n$. and so on.

Let us call conditions 1, 2, etc. the convexity conditions.

Let $C_n$ be the subset of $X_n$ consisting of all collections $(v_{ij})$ arising from configurations $(\mathbf{x}_i)$, $1 \leq i \leq n$, of $n$ distinct points in $\mathbb{R}^3$, using the formulas:

$$v_{ij} = \frac{\mathbf{x}_j-\mathbf{x}_i}{|\mathbf{x}_j-\mathbf{x}_i|}$$

for $1 \leq i,j \leq n$, $i \neq j$. One can indeed check that elements of $C_n$ satisfy the convexity conditions above.

Let us call the closure $\bar{C}_n$ of $C_n$ the set of "geometric" collections of pairwise directions. My question can now be stated. Is every element of $X_n$ "geometric"? It is easy to show that $\bar{C}_n \subseteq X_n$. My question is whether or not $X_n$ is equal to $\bar{C}_n$.

Note: I have simplified my post, compared to previous versions, as this is the crucial point remaining. I apologize for that. I hope this shorter version would be easier to read.

  • $\begingroup$ Can you explain the need to take the closure $\bar C_n$? It's not obvious to me that $C_n$ itself is not closed. $\endgroup$ – Rahul Jul 13 at 6:37
  • 1
    $\begingroup$ @Rahul, suppose you have a configuration $\mathbf{x} = (\mathbf{x}_i)$ of $n$ distinct points, and let two of the points collide, keeping track of the direction of their collision, so to speak. This can be done in a compactification of the configuration space. The collection of pairwise directions of such a "degenerate" configuration is in the closure of $C_n$, without being in $C_n$ itself. $\endgroup$ – Malkoun Jul 13 at 6:41

I have a somewhat complicated argument by contradiction that $X_n = \bar{C}_n$. Roughly speaking, here is an outline. Assume there is a point $v = (v_{ij}) \in X_n$ that is not in $\bar{C}_n$. Let $R$ be the set of all assignments $r = (r_{ij})$, with $r_{ij} \geq 0$ of "pairwise distances" such that

$$\max_{i\neq j} r_{ij} = 1.$$

Note that $R$ is compact. Given $r \in R$, and $1 \leq i<j<k \leq n$, define

$$\operatorname{hol}_{ijk}(v,r) = |r_{ij} v_{ij} + r_{jk} v_{jk} + r_{ki} v_{ki}|,$$

where $|\_|$ denotes the Euclidean norm in $\mathbb{R}^3$. Also define

$$\operatorname{hol}(v,r) = \operatorname{max}_{1 \leq i<j<k \leq n} \operatorname{hol}_{ijk}(v,r)$$

and call it the $2$-holonomy of $(v,r)$. Let

$$m = \operatorname{min}\{ \operatorname{hol}(v,r); r \in R \}$$

and let $r_{\min} \in R$ be an assignment of pairwise distances at which this minimum $m$ is attained. Note that $m>0$, otherwise $v$ would be geometric. We will construct, for the same $v$, an $r' \in R$, for which


thus reaching a contradiction.

Consider the set $S$ of all $(i,j,k)$, with $1 \leq i<j<k \leq n$, for which

$$\operatorname{hol}_{ijk}(v,r_{\min}) = \operatorname{hol}(v,r_{\min}).$$

We take each $(i,j,k) \in S$, and perturb the corresponding "distances" $(r_{\min})_{ij}$, $(r_{\min})_{jk}$, $(r_{\min})_{ki}$, so that $\operatorname{hol}_{ijk}(v,r')$ is smaller than the original value of $\operatorname{hol}_{ijk}(v,r_{\min})$, and we do that carefully while respecting that the $\max_{i\neq j} r'_{ij}$ is $1$. This requires considering several cases, depending on whether the number of $1$s among the values of the $3$ pairwise distances is $0$, $1$, $2$ or $3$.

After at most $|S|$ steps, we are guaranteed to have lowered the $2$-holonomy, while respecting that $\max_{i \neq j} r'_{ij} = 1$. We have thus reached a contradiction, and proved that $X_n = \bar{C}_n$.

I apologize for having skipped the case-by-case analysis step, but it should be easy to fill out the blanks by an interested reader.

Observation: interestingly, we only used conditions $1$ and $2$ ($2$-convexity and $3$-convexity respectively) in our proof. Thus the "higher" convexity conditions turn out to be redundant, in that $X_n$ could have been defined using only conditions $1$ and $2$.


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