Hidden convexity in eigenvalue problems, Fisher Information and Picone-type Inequalities

We show that some well-known convexity principles are equivalent to a geodesic convexity of Dirichlet integrals with applications in non-linear eigenvalue problems. We survey the link with an elementary inequality named after Picone, useful to get Hardy-type inequalities and similar estimates. The results are obtained in collaboration with Lorenzo Brasco.

Keywords: Uniqueness of eigenfunctions, convexity of Information functionals, Hardy inequalities