  We study the isoperimetric subgraphs of the infinite cluster
$\textbf{C}_\infty$ for supercritical bond percolation on $\mathbb{Z}^d$ with
$d\geq 3$. Specifically, we consider the subgraphs of $\textbf{C}_\infty \cap
[-n,n]^d$ which have minimal open edge boundary to volume ratio. We prove a
shape theorem for these subgraphs, obtaining that when suitably rescaled, these
subgraphs converge almost surely to a translate of a deterministic shape. This
deterministic shape is itself an isoperimetric set for a norm we construct. As
a corollary, we obtain sharp asymptotics on a natural modification of the
Cheeger constant for $\textbf{C}_\infty \cap [-n,n]^d$. This settles a
conjecture of Benjamini for the version of the Cheeger constant defined here.
