We consider an isoperimetric inequality for $(m+1)$-dimensional area minimizing submanifolds of arbitrary codimension which lie outside a convex set $\mathcal{ K} \subset \mathbb{ R}^{ n+1}$ and are bounded by a submanifold of $\mathbb{ R}^{ n+1} \setminus \mathcal{ K}$ and the convex set $\mathcal{ K}$. We show that the least value of the isoperimetric ratio is attained for an $(m+1)$-dimensional flat half-disk of $\mathbb{ R}^{ n+1}_+$. This extends prior work of Choe, Ghomi, and Ritor\'{ e} in codimension one and proves a conjecture of Choe in the case of relative area minimizers.
Categories: math.OC