3 problems

Let $(X,d,\mu)$ be a measured Gromov--Hausdorff limit of $n$-dimensional compact Riemannian manifolds $M_i$ with $\operatorname{Ric}_{M_i}\ge (n-1)k$, equipped with normalized Riem…

Let $\mathbb{H}^3 \times \mathbb{R}^2$ carry the product metric described in the example, and let $\operatorname{volume}(t)$ denote the volume of a geodesic ball of radius $t$. Let…

Let $\mathbb{S}^n$ be the sphere with its canonical Riemannian geometry, and let $o\in\mathbb{S}^n$. Let $X\subset\mathbb{S}^n$ be convex with $o$ in its interior, and let…