Titolo della tesi: Modern aspects of Convexity and the interplay between Geometry and Analysis
This thesis focuses on three main topics: First variations for measures of epigraphs of convex functions, Valuations on convex functions, and symmetrization processes. Our investigation approach is based mainly on the geometric properties of the entities at hand, aiming to recover analytic results. Expanding on the modern literature, we obtain new results in the fields mentioned above and present new points of view which might lead to further developments.