  A family of subsets of $\{1,2,\ldots,n\}$ is said to be {\em antipodal} if it
is closed under taking complements. We prove a best-possible isoperimetric
inequality for antipodal families of subsets of $\{1,2,\ldots,n\}$. Our
inequality implies that for any $k \in \mathbb{N}$, among all such families of
size $2^k$, a family consisting of the union of a $(k-1)$-dimensional subcube
and its antipode has the smallest possible edge boundary.
