Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.