Page 1 of 1

### 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 $$\mathbb{S}^{m-1}$$?

### Re: Question on a metric

Posted: Tue Sep 27, 2016 8:24 pm
Let $$\displaystyle{x\,,y\in S^{m-1}}$$ such that $$\displaystyle{d(x,y)=\theta}$$.

Consider the perpendicular to $$\displaystyle{xy}$$ - line from the origin. Then,

$$\displaystyle{\sin\,\dfrac{\theta}{2}=\dfrac{||x-y||_{2}}{2}\iff ||x-y||_{2}=2\,\sin\,\dfrac{\theta}{2}}$$.

Yes, the function $$\displaystyle{d}$$ is a metric in $$\displaystyle{S^{m-1}}$$ and with this metric,

the triplet $$\displaystyle{\left(S^{m-1}\,,d\,,\sigma\right)}$$, where,

$$\displaystyle{\sigma(A)=\dfrac{|\left\{s\,x\in \mathbb{R}^m\,,0\leq s\leq 1\right\}|}{|B_{2}^m|}\,,\forall\,A\in\mathbb{B}(S^{m-1})}$$,

becomes a metric probability space.