Question on a metric
Posted: Thu Jul 14, 2016 1:40 pm
Consider the unit Eucleidian sphere \( \mathbb{S}^{m-1}=\left \{ x \in \mathbb{R}^m :\left \| x \right \|_2=1 \right \} \) in \( \mathbb{R}^m \). We will define "distance" \( d(x, y) \) of two points \( x, y \in \mathbb{S}^{m-1} \) to be the convex angle \( xO y \) that is defined by the origin and the \( x, y \) points.
a) Show that if \( d\left ( x, y \right )=\theta \) then \( \displaystyle \left \| x-y \right \|_2=2\sin \frac{\theta }{2} \) holds.
b) Is \( d \) a metric in \( \mathbb{S}^{m-1} \)?
a) Show that if \( d\left ( x, y \right )=\theta \) then \( \displaystyle \left \| x-y \right \|_2=2\sin \frac{\theta }{2} \) holds.
b) Is \( d \) a metric in \( \mathbb{S}^{m-1} \)?