Smooth Approximations of Quasispheres

We prove that every $n$-dimensional quasisphere is the Gromov-Hausdorff limit of a sequence of locally smooth uniform quasispheres.

We also prove an analogous result in the bi-Lipschitz setting.

This extends recent results of D.

Ntalampekos from dimension 2 to arbitrary dimension.

In the process, we replace the second half of his argument by a completely different, more efficient approach, which should be applicable to other problems.