# Distance function-inequality in unit sphere

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 $$f(x)=f(-x)$$.

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