  We prove that the principal eigenvalue of any fully nonlinear homogeneous
elliptic operator which fulfills a very simple convexity assumption satisfies a
Brunn-Minkowski type inequality on the class of open bounded sets in
$\mathbb{R}^n$ satisfying a uniform exterior sphere condition. In particular
the result applies to the (possibly normalized) $p$-Laplacian, and to the
minimal Pucci operator. The proof is inspired by the approach introduced by
Colesanti for the principal frequency of the Laplacian within the class of
convex domains, and relies on a generalization of the convex envelope method by
Alvarez-Lasry-Lions. We also deal with the existence and log-concavity of
positive viscosity eigenfunctions.

Analysis of PDEs