  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 $8$.
