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

The Brunn-Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators