Abstract: The classical notion of convexity of subsets of the Euclidean space metamorphoses into several distinct types of (geodesic) convexity in the more general Riemannian setting , this phenomenon, along with its geometric implications, being the main topic of this talk.

Abstract: The circle is uniquely characterized by the property that among all simple closed plane curves of given length L, the circle of circumference L encloses maximum area. In this talk, we will recount some geometric generalizations and applications of this inequality. We shall also consider various geometric and analytic inequalities closely connected to the isoperimetric inequality, such as Brunn-Minkowski's, Wirtinger's, Poincaré's and Sobolev's inequalities.