The normalised volume of toric singularities

In my talk I will discuss the notion of normalised volume for log terminal singularities. For the special case of toric singularities this turns out to be closely related to the notions of Mahler volume and Santaló point in convex geometry. I will explain how well-known facts from convex geometry can be utilised to deduce non-trivial statements about toric singularities. The Mahler conjecture might have a short appearance as well.