On the one hand, methods and ideas of convex analysis are beneficial in many different areas of mathematics and physics, starting with functional analysis, optimal control theory and quantum mechanics. On the other hand, the basic facts in convexity are quite simple and usually have a nice geometric interpretation. This course aims to introduce several basic questions regarding the distribution of volume in convex bodies from a geometric and analytic point of view. The precise choice of topics will depend on the audience's background; sample topics include - Brunn-Minkowski inequality and its applications - Marginal of convex sets, classes of concave functions - Symmetrizations and shadow movements - Duality and Santalo’s inequality.