AbstractThe variational capacity {\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the
density dichotomy at large scales, namely that for every {E\subset{\mathbb{R}}^{n}},\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}%
{\operatorname{cap}_{p}(B(x,r),B(x,2r))}is either zero or tends to 1 as {r\to\infty}.
We prove that this property still holds in unbounded
complete geodesic metric spaces equipped
with a doubling measure supporting a p-Poincaré inequality,
but that it can fail in nongeodesic metric spaces and also for the Sobolev
capacity in {{\mathbb{R}}^{n}}.
It turns out that
the shape of balls impacts the validity of the density dichotomy.
Even in more general metric spaces, we construct families of sets, such as
John domains, for which the density dichotomy holds.
Our arguments include an exact formula for the variational capacity of
superlevel sets for capacitary potentials
and a quantitative approximation from inside of
the variational capacity.
A doubling measure on $\mathbb {R}^d$ can charge a rectifiable curve
