Let $\mathbb{S}$ be the unit sphere, and let $v_1\in\mathbb{R}^d$ be the north-pole. Let $\mathcal{E}$ be a symmetric convex body (with respect to the geodesics on the sphere) and let the minkowski functional w.r.t. $\mathcal E$ be defined as follows: For any vector $y\in\mathbb{R}^d$, $||y-v_1||_{\mathcal{E}}=\inf_r\{r>0:(y-v_1)\in r\mathcal{E}\}$, i.e., the smallest geodesic stretch required for $\mathcal{E}$ centered at $v_1$ to cover $y$. My question is the following: For any two points $v_1$ and $v_2$ on the hemisphere, does triangle inequality hold, i.e., is $\left|||y-v_1||_\mathcal{E}-||y-v_2||_\mathcal{E}\right|\leq\|v_1-v_2\|_\mathcal{E}$ true ? Can someone point me to a relevant literature?

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