The Isoperimetric Inequality for Compact Rank One Symmetric Spaces and Beyond

Abstract: Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the classical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the real projective space $\mathbb{R}P^n$, we demonstrate that the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{R}P^k\subset\mathbb{R}P^n$. For the complex projective space $\mathbb{C}P^n$, the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{C}P^k\subset\mathbb{C}P^n$. And for the quaternionic projective space, the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{H}P^k\subset\mathbb{H}P^n$.