Hyperspherical radius is hard to approximate
For a closed Riemannian (or just metric) manifold $M^n$, the hyperspherical radius is the largest radius $r$ such that $M$ has a contracting degree 1 map to the sphere of radius $r$. For a metric on the sphere, this provides a possible measure of "roundness"; for a sphere of bounded local geometry, the hyperspherical radius must be roughly between 1 (for a perfectly skinny sphere) and the $n$th root of the volume (for a perfectly round one). For $n=2$, Guth gave easily computable bounds for the hyperspherical radius. When $n \geq 3$, it turns out that, given a simplicial complex homeomorphic to the sphere, it's NP hard to pin down the hyperspherical radius much more than this. This provides some rigorous support for the observation that it's quite hard to prove general results about metrics on the 3-sphere, and is joint work with Zarathustra Brady and Larry Guth.