Sorry if this is an obvious question but I have not been able to find a straightforward answer for it. My intuition tells me this is correct, and have confirmed that the scipy convex hull algorithm will return all samples used to generate a Fibonacci sphere as hull vertices.
I am not a math person, but a coder trying to test the performance scaling of a script, so if someone could confirm this (with a proof if you're feeling generous) I would much appreciate it. Thank you.
Recall these two pertinent facts about the norm/length function: \begin{align*} \|x + y\| &\le \|x\| + \|y\| \\ \|\lambda x\| &= |\lambda| \|x\|, \end{align*} where $x$ and $y$ are vectors and $\lambda$ is a scalar.
The unit ball is defined by $\{x : \|x\| \le 1\}$. Suppose $x$ and $y$ lie in this ball. We wish to show that the line segment between $x$ and $y$, i.e. all points of the form $\lambda x + (1 - \lambda) y$ where $\lambda \in [0, 1]$, also lies in the ball. We have, \begin{align*} \|\lambda x + (1 - \lambda)y\| &\le \|\lambda x\| + \|(1 - \lambda)y\| \\ &=|\lambda| \|x\| + |1 - \lambda|\|y\| \\ &\le |\lambda| + |1 - \lambda| \end{align*} When $\lambda \in [0, 1]$, then $\lambda, 1 - \lambda \ge 0$, so $$\|\lambda x + (1 - \lambda)y\| \le \lambda + 1 - \lambda = 1,$$ which is what we wanted to show.