On the packing dimension of distance sets with respect to $C^1$ and polyhedral norms

We prove that, for every polyhedral or $C^1$ norm on ${\mathbb{R}}^d$ and every set $E\subseteq {\mathbb{R}}^d$ of packing dimension $s$, the packing dimension of the distance set of $E$ with respect to that norm is at least $\frac{s}{d}$. One of the main tools is a nonlinear projection theorem extending a result of M. Järvenpää. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms.