  Erd\H{o}s' unit distance problem and Erd\H{o}s' distinct distances problem
are among the most classical and well-known open problems in all of discrete
mathematics. They ask for the maximum number of unit distances, or the minimum
number of distinct distances, respectively, determined by $n$ points in the
Euclidean plane. The question of what happens in these problems if one
considers normed spaces other than the Euclidean plane has been raised in the
1980s by Ulam and Erd\H{o}s and attracted a lot of attention over the years. We
give an essentially tight answer to both questions for almost all norms on
$\mathbb{R}^d$, in a certain Baire categoric sense.
  For the unit distance problem we prove that for almost all norms ||.|| on
$\mathbb{R}^d$, any set of $n$ points defines at most $\frac{1}{2} d \cdot n
\log_2 n$ unit distances according to ||.||. We also show that this is
essentially tight, by proving that for every norm ||.|| on $\mathbb{R}^d$, for
any large $n$, we can find $n$ points defining at least
$\frac{1}{2}(d-1-o(1))\cdot n \log_2 n$ unit distances according to ||.||.
  For the distinct distances problem, we prove that for almost all norms ||.||
on $\mathbb{R}^d$ any set of $n$ points defines at least $(1-o(1))n$ distinct
distances according to ||.||. This is clearly tight up to the $o(1)$ term.
  Our results settle, in a strong and somewhat surprising form, problems and
conjectures of Brass, of Matou\v{s}ek, and of Brass-Moser-Pach. The proofs
combine combinatorial and geometric ideas with tools from Linear Algebra,
Topology and Algebraic Geometry.
