We study the spectral convergence of compact Riemannian manifolds in relation with the Gromov-Hausdorff distance and discuss the geodesic distances and the energy forms of the limit spaces.
These are the lecture notes of a minicourse given at a winter school in Marseille 2011. The aim of the course was to give an introduction to recent work on the geometry of the space of Kähler metrics associated to an ample line bundle. The emphasis of the course was the role of convexity, both as a motivating example and as a tool.