Consider a vector $v$ and a collection of vectors $u_1, \dots, u_n$, such that for each individual vector $u_i$ we have $\angle(v,u_i) \leq \frac{\pi}{4}$.
Now consider a vector $w = c_1 \cdot u_i + c_2 \cdot u_j$ with $c_1,c_2 > 0$ for two $i,j \in \{1, \dots, n\}$ ($w$ is a positive linear combination of two vectors in $u_1, \dots, u_n$.
Does $\angle(v, w) \leq \frac{\pi}{4}$ hold?
It feels like that is true, but I have no clue how to prove it.
Best Answer
We have that
$$\angle(v,u_i) \leq \frac{\pi}{4} \iff \frac{v\cdot u_k}{|v||u_k|}\ge \frac {\sqrt 2} 2 \iff v\cdot u_k\ge \frac {\sqrt 2} 2|v||u_k|$$
and we need to check that
$$\frac{v\cdot w}{|v||w|}\ge \frac {\sqrt 2} 2\iff v \cdot (c_1u_i+c_2u_j) \ge \frac {\sqrt 2} 2|v||c_1u_i+c_2u_j|$$
which is true indeed
$$v \cdot (c_1u_i+c_2u_j) =c_1v\cdot u_i + c_2v\cdot u_j \ge \frac {\sqrt 2} 2|v|\left(c_1|u_i|+c_2|u_j|\right)$$
and by triangle inequality
$$c_1|u_i|+c_2|u_j|\ge |c_1u_i+c_2u_j|$$