Rigidity and large volume residues in exterior isoperimetry for convex sets

A comparison theorem by Choe, Ghomi and Ritoré states that the exterior isoperimetric profile IC of any convex body C in RN lies above that of any half-space H. We characterize convex bodies such that IC≡IH in terms of a notion of “maximal affine dimension at infinity”, briefly called the asymptotic dimension d⁎(C) of C. More precisely, we show that IC≡IH if and only if d⁎(C)≥N−1. We also show that if d⁎(C)≤N−2, then, for large volumes, IC is asymptotic to the isoperimetric profile of RN. We then estimate, in terms of d⁎(C)-dependent power laws, the order as v→∞ of the difference between IC and the isoperimetric profile of RN.