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For $1 \le p \le q \le \infty$, I need an inequality bounding the $\ell_q$ norm from above by the $\ell_p$ norm on $\mathbb{R}^n$: finding a $\lambda$ so that $$ \Vert v \Vert_q \le \lambda \Vert v \V …

There are lots of ways to do this; without more constraints (or indication of what is desired), it's hard to pick a best one. But here's one. We'll exploit the fact that for any two triangles, there …

This is expanding on @user35593 's comment above. Let $\bar{C}$ be the closure of the convex hull of $x_0,\dots,x_n$, and let $x'$ be the projection of $x$ onto $\bar{C}$. By Bridson and Haefliger, P …

(Split off from Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees? ) Fix an integer $k \ge 2$, and let $MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squa …

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) , I started wondering if the center of gravity is always contained in the closed convex hull. More precisely, given $ …