  We study range spaces, where the ground set consists of either polygonal
curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain
holes and the ranges are balls defined by an elastic distance measure, such as
the Hausdorff distance, the Fr\'echet distance and the dynamic time warping
distance. The range spaces appear in various applications like classification,
range counting, density estimation and clustering when the instances are
trajectories, time series or polygons. The Vapnik-Chervonenkis dimension
(VC-dimension) plays an important role when designing algorithms for these
range spaces. We show for the Fr\'echet distance of polygonal curves and the
Hausdorff distance of polygonal curves and planar polygonal regions that the
VC-dimension is upper-bounded by $O(dk\log(km))$ where $k$ is the complexity of
the center of a ball, $m$ is the complexity of the polygonal curve or region in
the ground set, and $d$ is the ambient dimension. For $d \geq 4$ this bound is
tight in each of the parameters $d, k$ and $m$ separately. For the dynamic time
warping distance of polygonal curves, our analysis directly yields an
upper-bound of $O(\min(dk^2\log(m),dkm\log(k)))$.
