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I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space: The first two give the torus and the Klein bottle, respectively. What are the …

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees? Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ t …

Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the s …

(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 …

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 …

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 $ …

I have a family of possibly related questions. Let me start with an elementary one: Question 1. Fix an integer $n$. For which collections of real numbers $a_{ij}$, $i, j = 1, \dots, n$, is it true th …

I just came across this old question, which I happened to think about earlier. Let me give a more explicit description. As you noted, the binary tetrahedral group forms the vertices of a 24-cell. The …

They are honestly used all over the place, it's easy for this to get out of control. Maybe one non-obvious place is the construction of the canonical triangulation of a hyperbolic 3-manifold: Epstein …