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 $\tfrac{ s}{ d}$. One of the main tools is a nonlinear projection theorem extending a result of M. J\"{ a}rvenp\"{ a}\"{ a}. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms.
Categories: math.CA