Rigidity and large volume residues in exterior isoperimetry for convex sets

A comparison theorem by Choe, Ghomi and Ritor\'e states that the exterior isoperimetric profile $I_\mathcal{C}$ of any convex body $\mathcal{C}$ in $\mathbb{R}^N$ lies above that of any half-space $H$. We characterize convex bodies such that $I_\mathcal{C}\equiv I_H$ in terms of a notion of "maximal affine dimension at infinity'', briefly called the asymptotic dimension $d^*(\mathcal{C})$ of $\mathcal{C}$. More precisely, we show that $I_\mathcal{C}\equiv I_H$ if and only if $d^*(\mathcal{C})\ge N-1$. We also show that if $d^*(\mathcal{C})\le N-2$, then, for large volumes, $I_\mathcal{C}$ is asymptotic to the isoperimetric profile of $\mathbb{R}^N$. We then estimate, in terms of $d^*(\mathcal{C})$-dependent power laws, the order as $v\to\infty$ of the difference between $I_\mathcal{C}$ and the isoperimetric profile of $\mathbb{R}^N$.