Suppose I have a great-circle of a sphere in $\mathbb{R}^n$, the chord length (the euclidean distance of any two points) is $L$, how can we upper bound the arc length $C$ (for any radius)?
I read somewhere $C\le \frac{\pi L}{2}$, is it correct? If so, how to prove it?
Thanks.
If the longer of the two arcs is allowed, then there is no bound: An arbitrarily short chord can correspond to an arbitrarily long arc.
Otherwise, let $\alpha\le \frac\pi2$ be half the subtending angle. Then the arc length is $C=2r\alpha$ and the chord length is $L=2r\sin\alpha$. Hence $\frac CL=\frac{\alpha}{\sin\alpha}$. This expression grows with $\alpha$ (tends to $1$ as $\alpha\to 0$), hence has its maximum at the right end, i.e., for $\alpha=\frac\pi2$, where the quotient is $\frac\pi2$.