> Some geometric inequalities such as the classical isoperimetric > inequality are considerably easier to prove in two than in higher dimensions. > In my lecture I will present two long conjectured and recently > confirmed inequalities, for which already the proof in two dimensions > presents a considerable challenge. > The first one involves the elastic energy $\int_\gamma \kappa^2\,ds$ > of a plane closed curve $\gamma$ bounding a simply connected set $\Omega$. > Given the area of $\Omega$, only the circle minimizes the elastic > energy of $\Omega$. > The second one relates the length of an area-bisecting shortest curve > to the shape of the admissible convex plane set $\Omega$ of given area. > Only the disc maximizes the length of this curve among all plane > convex sets of given area.