Algebraic Geometry is (or used to be) about solving polynomial equations. An important example is the Riemann-Roch problem: given a compact Riemann surface $\Sigma$, find the dimension of the vector space of rational functions on $\Sigma$ with predescribed poles. Generalizing this problem suitably to higher dimensions leads to the notion of volume of a divisor on a variety, the focus of much recent research activity.
It is well known that there is a dictionary between convex bodies and (toric) algebraic geometry. In particular, Teissier and Khovanskii showed that the Alexandrov-Fenchel inequalities can be derived from the Hodge Inequalities. The talk will be a non-technical survey of this circle of ideas, covering mostly classical material.