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Assume that $d$ is a distance function (i.e. Riemannian metric) on a unit sphere $X$ of Euclidean space $\mathbb{E}^3$ (That is, $d$ is an angle between two unit vectors). If $e_i$ is an orthonormal basis in $\mathbb{E}^3$, i.e. $d(e_i,e_j)=\frac{\pi}{2}$ for all $i\neq j$, then prove that

$$ \sum_{i=1}^3 \ \bigg|\pi - d(e_i,u)-d(e_i,v) \bigg| \geq \pi - d(u,v) $$

Proof : When $f(x)=\bigg|\pi - T(x)\bigg|$ and $T(x)=d(x,u)+d(x,v)$, then $T$ is a 2-Lipschitz function. And $f(x)=f(-x)$.

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  • $\begingroup$ What's $S^2(1)$? $\endgroup$ – Alex R. Apr 17 '18 at 3:51
  • $\begingroup$ It is a unit sphere in $\mathbb{R}^3$ and $d$ is intrinsic metric on it. $\endgroup$ – HK Lee Apr 17 '18 at 3:52
  • $\begingroup$ Are u,v generic vectors of $\mathbb{R}^3$? $\endgroup$ – big-lion Apr 17 '18 at 3:59
  • $\begingroup$ @big-lion : I edited. $\endgroup$ – HK Lee Apr 17 '18 at 4:03

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