VC dimension of only the rim of a unit disk
Let us prove the following general result:


  Let $\mathcal F$ be a class of functions from $\mathcal X$ to $\{0,1\}$. If $\mathcal F$ has VC dimension $d$ then $|\mathcal F| \geq 2^d$.


Indeed, if $\mathcal F$ has VC dimension $d$ then $\mathcal F$ shatters some set $S \subseteq \mathcal X$ of size $d$. This means that for any function $\phi\colon S \to \{0,1\}$ we can find a function $f_\phi \in \mathcal{F}$ that agrees with $\phi$ on $S$. Note that if $\phi \neq \psi$ then $f_\phi \neq f_\psi$. Since there are $2^d$ different functions from $S$ to $\{0,1\}$, we conclude that $\mathcal F$ must contain at least $2^d$ functions. $\square$

In particular, the VC dimension of a class of functions containing a single function is zero.