# Is the set of collections of directions satisfying the following convexity conditions “geometric”?

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.

• 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. – Rahul Jul 13 at 6:37
• @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. – 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, 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

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

$$\operatorname{hol}(v,r')<\operatorname{hol}(v,r_{\min})$$

Consider the set $$S$$ of all $$(i,j,k)$$, with $$1 \leq i, for which
$$\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$$.
Observation: interestingly, we only used conditions $$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$$.