VC dimension upper bound for Euclidean balls
= VC dimension upper bound for Euclidean balls
{c}
The indicators of closed <Euclidean ball>[Euclidean balls] in $\mathbb R^d$ have VC dimension at most $d+2$. Indeed,
$$
\mathbf1_{\{\lVert x-c\rVert_2^2\leq r^2\}}
=\mathbf1_{\{f_{c,r}(x)\leq0\}},
$$
where
$$
f_{c,r}(x)=\lVert x\rVert_2^2-2c^Tx+\lVert c\rVert_2^2-r^2
$$
belongs to the span of $\lVert x\rVert_2^2,x_1,\ldots,x_d,1$.