Setting:

Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,\dotsc,n$. Let $a \in R^d$ be a unit vector.

Goal:

We want to show that the following hold or to find a counter-example.

$$\sum_{i=1}^n \lvert u_i^T a\rvert u_i^T w < 1.$$

We can prove this statement for the case of $n=2$, but we were not able to prove it for $n>2$.

Any suggestions?