Rigidity and large volume residues in exterior isoperimetry for convex sets

Abstract

A comparison theorem by Choe, Ghomi and Ritoré 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 $d^{\ast }(\mathcal{C})$ of $\mathcal{C}$. More precisely, we show that $I_{\mathcal{C}}\equiv I_H$ if and only if $d^{\ast }(\mathcal{C})\ge N-1$. We also show that if $d^{\ast }(\mathcal{C})\le N-2$, then, for large volumes, $I_{\mathcal{C}}$ is asymptotic to the isoperimetric profile of ${\mathbb{R}}^N$. We then estimate, in terms of $d^{\ast }(\mathcal{C})$-dependent power laws, the order as $v\to \infty$ of the difference between $I_{\mathcal{C}}$ and the isoperimetric profile of ${\mathbb{R}}^N$.