The Isoperimetric Inequality for Compact Rank One Symmetric Spaces and Beyond

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 $\mathbb{H}P^k\subset\mathbb{H}P^n$.