Modern aspects of convexity and the interplay between geometry and analysis

In this thesis, we explore the interactions between geometry and analysis in the world of convex geometry. In particular, we mix techniques from the world of convex and functional analysis together with some from non-smooth differential geometry, finding new interpretations of various phenomena appearing in recent developments in convexity. This work has three main focuses: First variations for volumes of convex functions, characterization results for valuations on convex functions, and convergence properties of symmerizations processes.