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## Question on a metric

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Tolaso J Kos
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### Question on a metric

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 $\mathbb{S}^{m-1}$?
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Papapetros Vaggelis
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### Re: Question on a metric

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.