The convex hull of compact adjoint orbits

The Satake–Furstenberg compactification of a real symmetric space of non-compact type is closely related to the convex hull $\hat{\mathrm O}$ of a $K$-orbit $\mathrm O$ in $\mathfrak p$, where $\mathfrak g=\mathfrak k+\mathfrak p$ denotes a Cartan decomposition of the Lie algebra of a semisimple Lie group $G$. In this talk we give a precise description of $\hat{\mathrm{O}}$ in terms of the geometry of $\mathrm O$ given by compact orbits of minimal parabolic subgroups of $G$ in $\mathrm O$.