Random rotation of a set of distinct points in $\mathbb{R}^n$ [closed]

Question

Consider a set $\{\mathbf{X}_1,\cdots , \mathbf{X}_M\}$ of distinct points in $\mathbb{R}^n$ with $M$ finite. The $M$ values of the $i$-th coordinate do not all have to be dinstinct. For example, in $\mathbb{R}^2$ the points could be placed on a regular grid so that several points have the same x or y coordinates.

Let's now consider a rotation, that maps the original set into a new set $\{\mathbf{X}_1', \cdots, \mathbf{X}_M'\}$.

I would like to show that if we select the rotation randomly, i.e. if we project on a random orthogonal basis, the $M$ values of the $i$-th coordinates of the new set will be distinct for all $i=1, \cdots, n$ with probability almost one.

Intuitively it sounds like that should be the case. But I have no experience with random matrices and I wonder where I could start from to prove it.

Answer 1

For any two distinct integers $l,m \in \{1,\ldots, M\}$, let us define $\mathbb{P}_{lm}$; be the $(n-1)$-dimensional space consisting of vectors in $v \in \mathbb{R}_n$ satisfying $v^T(\mathbf{X}_l-\mathbf{X}_m) = 0$ Then as long as $\mathbf{X}_l-\mathbf{X}_m$ is nonzero the probability that a randomly chosen vector of norm 1 from ${\mathbb{R}}_n$ lands in $\mathbb{P}_{lm}$ is 0 (as it would be for any $(n-1)$-dimensional space).

So let $A$ be the randomly chosen matrix from $SO_3$. For any positive integers $l,m \le M$ and $p \le n$ only way $A\mathbf{X}_l$ and $A\mathbf{X}_m$ can agree on the $p$-th coordinate is if the $p$-th row $v_p$ of $A$ is in $\mathbb{P}_{lm}$. However, given $A$ randomly chosen from $SO_3$ according to the uniform distribution, then for any one such integer $p$, the $p$-th row $v_p$ of $A$ is also chosen according to the uniform distribution from the set of vectors of norm 1. The probability of $v_p$ so chosen being in $\mathbb{P}_{lm}$ is 0. Thus, for any one choice of $p,l,m$ the probability that the $p$-th coordinate of $A\mathbf{X}_l$ equals the $p$-th coordinate of $\mathbf{X}_m$ is 0. To conclude that the probability is 0 that there is any such $p,l,m$ where the $p$-th coordinate of $A\mathbf{X}_l$ equals the $p$-th coordinate of $\mathbf{X}_m$ use the Union Bound; only $nM^2$ such choices for $p,l,m$.