On a isoperimetric-isodiametric inequality

The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed isoperimetric-isodiametric inequalities in Riemannian manifolds. We first prove that the same inequality, with the sharp Euclidean constants, holds on Cartan-Hadamard spaces as well as on minimal submanifolds of ${\mathbb{R}}^n$. The equality cases are also studied and completely characterized; in particular, the latter gives a new link with free boundary minimal submanifolds in a Euclidean ball. We also consider the case of manifolds with non-negative Ricci curvature and prove a new comparison result stating that metric balls in the manifold have product of boundary area and radius bounded by the Euclidean counterpart and equality holds if and only if the ball is actually Euclidean. We then pass to consider the problem of the existence and the regularity of optimal shapes (i.e. regions minimizing the product of boundary area and radius under the constraint of having fixed enclosed volume), called here isoperimetric-isodiametric regions. We give examples of spaces where there exists no isoperimetric-isodiametric region and we prove that on the other hand on compact spaces and on $C^0$-locally-asymptotic Euclidean manifolds with non-negative Ricci curvature there exists an isoperimetric-isodiametric region for every positive volume (this last class of spaces includes a large family of metrics playing a key role in general relativity and Ricci flow: the so called Hawking gravitational instantons and the Bryant-type Ricci solitons). We finally prove that the boundary of the isoperimetric-isodiametric regions satisfy the optimal $C^{1,1}$ regularity outside of a closed subset of Hausdorff co-dimension $8$.