  Weak and strong coloring numbers are generalizations of the degeneracy of a
graph, where for each natural number $k$, we seek a vertex ordering such every
vertex can (weakly respectively strongly) reach in $k$ steps only few vertices
with lower index in the ordering. Both notions capture the sparsity of a graph
or a graph class, and have interesting applications in the structural and
algorithmic graph theory. Recently, the first author together with McCarty and
Norin observed a natural volume-based upper bound for the strong coloring
numbers of intersection graphs of well-behaved objects in $\mathbb{R}^d$, such
as homothets of a centrally symmetric compact convex object, or comparable
axis-aligned boxes.
  In this paper, we prove upper and lower bounds for the $k$-th weak coloring
numbers of these classes of intersection graphs. As a consequence, we describe
a natural graph class whose strong coloring numbers are polynomial in $k$, but
the weak coloring numbers are exponential. We also observe a surprising
difference in terms of the dependence of the weak coloring numbers on the
dimension between touching graphs of balls (single-exponential) and hypercubes
(double-exponential).
