We define the unit circle as the collection of all vectors with length 1 centered at some point. (The one below specifically defines the unit circle centered at the origin)
$$ \mathscr{C}_2 = \{x\in\mathbb{R}^2: ||x||_2=1\} $$
In this case, the norm used is the Euclidean norm $p=2$. But we may use the $\ell_p$ metric for $p\in[1,\infty)$ where
$$ ||x||_p = \left(|x_1|^p + |x_2|^p \right)^{1/p} $$
We use these metrics to generalize the unit circle as the $\ell_p$ unit circles: $$ \mathscr{C}_p = \{x\in\mathbb{R}^2: ||x||_p=1\} $$
$\ell_p$ Unit Circles" />These circles have their varying circumferences, and $\pi_p$ generalizes our notion of the ratio between a circle's circumference to its diameter. Richter provides generalized trigonometric functions for $\ell_p$ metric spaces. Specifically,
$$\cos_p(\theta) = \frac{\cos(\theta)}{N_p(\theta)}$$
$$\sin_p(\theta) = \frac{\sin(\theta)}{N_p(\theta)}$$
where, $N_p(\theta) = \left(\|\cos(\theta)\|^p + \|\sin(\theta)\|^p\right)^{1/p}.$ These generalized trigonometric functions also parametrize the curve of $\mathscr{C}_p$ as $r_p(\theta) = (\cos_p(\theta), \sin_p(\theta))$.
I seek to prove that for the generalized $\ell_p$ circle, that as the arc length of $V\subset\mathscr{C}_p$ increases from $0$ to $\pi_p$ so too its chord. However, this is not necessarily true for all arcs $V$. Observe for the $\ell_1$ case, and consider points on $\mathscr{C}_p$, $A=(\frac{1}{2}, \frac{1}{2})$, $B=(-\frac{1}{2},\frac{1}{2})$, $C=(1,0)$ and $D=(0,1)$. The arcs $AB$ and $CD$ have the same arc distance, but the chords between them do not have the same length. Thus, it was useful to come up with the concept of an arc's orientation around $\mathscr{C}_p$. We designate the tangential angle $\theta$ such that point $M=r_p(\theta)$ and $M$ splits the arc into two equal lengths. By the limited symmetries of $\mathscr{C}_p$, we say that $\theta\in[0,\pi/4]$ and it represents an arc's orientation.
What I want to show is, will increasing an arcs length from $u$ to $v$ while staying fixed at some $\theta$ always increase the corresponding chord's length? In fact, I know this to be true via numerical verification but I want to prove it.
I came across the Hinge Theorem and I want to see if it can be generalized to non-Euclidean metrics. It's page on Wikipedia states that "The hinge theorem holds in Euclidean spaces and more generally in simply connected non-positively curved space forms." However, no reference is given. Is this claim true? How would I go about solving it?
I attempted to reconstruct the proof technique but it requires rotating line segments/arcs around and this may not hold. Any help would greatly be appreciated.