An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space

We construct families of smooth functions $H\text{\hspace{0.17em}⁣}:{\mathbb{R}}^{n+1}\to \mathbb{R}$ such that the Euclidean $(n+1)$-space is completely filled by not necessarily round hyperspheres of mean curvature $H$ at every point.