Given a set of N elements $X$ on the unit n-sphere $\mathbb{S}^n$ where $\forall \mathbf{x} \in X, ||\mathbf{x}||_2 = 1$. I would like to know if there is a minimum number of points required to guarantee:
$$ \sum_{\mathbf{x}_i, \mathbf{x}_j \in X i \neq j} \mathbf{x}_i^T \mathbf{x}_j \geq 0$$
My intuition is that for sufficiently many points in space, there is no configuration that has negative expected similarity as the expected maximum angle between points must decrease as a function of N.
No such $N$ exists, even for $n=1$: just take $N$ equally spaced points on the unit circle. For each fixed $i$, the sum over the remaining $j$ equals $$ \sum_{k=1}^{N-1} \cos\frac{2\pi k}N = -1. $$