The notion of mixed volumes and the inequalities involving them play a central role in the modern convex geometry, and have many connections to various other areas of mathematics. The most classical inequalities includes Brunn-Minkowski inequality and more general Alexandrov-Fenchel inequality.

In this talk we will discuss a local version of Alexandrov-Fenchel inequality and its connection to the search for the best constant in a number of geometric inequalities including inequality on the volume of orthogonal projections of convex bodies; isomorphic version of Bezout inequality; approximate submodularity of Minkowski sum; as well as to the property of distribution of roots of Steiner polynomial. This is a joint work with Matthieu Fradelizi and Mokshay Madiman.