The typical-norm distinct-distances conjecture for full-dimensional point sets

A $d$-norm is a norm $\|\cdot\|$ on $\mathbb{R}^d$, identified with its unit ball. A set of points is full-dimensional when it is not contained in an affine hyperplane, and a set of norms is meagre when it is a countable union of nowhere dense sets in the Hausdorff topology on unit balls. The typical-norm distinct-distances conjecture. For every fixed $d$, for all $d$-norms $\|\cdot\|$ outside a meagre set, and for all $n>n_0(d)$, every full-dimensional set of $n$ points in $\mathbb{R}^d$ determines at least

$$(d-o(1))n$$

distinct distances with respect to $\|\cdot\|$, where $o(1)\to 0$ as $n\to\infty$. The result is trivial for $d=1$ and open already for $d=2$; prior work gives only $(1-o(1))n$ distances for all norms outside a meagre set.