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Brian Krummel : Higher codimension relative isoperimetric inequality outside a convex set

We consider an isoperimetric inequality for area minimizing submanifolds $R$ lying outside a convex body $K$ in $\mathbb{R}^{n+1}$. Here $R$ is an $(m+1)$-dimensional submanifold whose boundary consists of a submanifold $T$ in $\mathbb{R}^{n+1} \setminus K$ and a free boundary (possibly not rectifiable) along $\partial K$. An isoperimetric inequality outside a convex body was previously proven by Choe, Ghomi, and Ritore in the codimension one setting where $m = n$. We extend their result to higher codimension. A key aspect of the proof are estimates on the concentration of mass of $T$ and $R$ near $\partial K$.

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