Let $p, x_i \in \mathbb{R^m}$, $ (i=1,..,n) $ with $||x_i||=||p||=1$ for all $i$.
Suppose that $p$ satisfies $$\sum_{i=1}^{n}\text{arccos}(\langle p,x_i\rangle) \:\leq\: \sum_{i=1}^{n}\text{arccos}(\langle u,x_i\rangle)$$ $\forall u\in \mathbb{R^m}, ||u||=1$
I believe that this implies the following inequality: $\sum_{i=1}^{n}\langle p,x_i\rangle \geq\sum_{i=1}^{n}\langle u,x_i\rangle \forall u\in \mathbb{R^m}, ||u||=1$
My reasons:
- $0 \leq |\langle p,x_i\rangle| \leq 1 $ (Cauchy-Schwarz inequality)
- $\text{arccos}:[-1,1] \rightarrow [0,\pi] $ is bijective and strictly decreasing
How do you prove this inequality? Or is there a contradiction?