Mixed volumes are known to have a wide range of applications in fields such as analysis, probability theory, algebraic combinatorics, and, more recently, the study of artificial neural networks (https://openreview.net/pdf?id=uREg3OHjLL). Our research focuses on identifying complete systems of inequalities for mixed volumes, as well as developing inequalities specific to mixed volumes of zonoids.

The problem of finding complete systems of inequalities dates back to the works of Heine (1938) and Shephard (1960). Recently, renewed interest in this problem has been sparked by the work of June Huhe and Petter Brändén on Lorentzian polynomials (https://arxiv.org/abs/1902.03719).

In this talk, I will present several inequalities obtained in collaboration with Katherina von Dichter, Richard Simon, and Ivans Soprunov (arXiv 2409.18928, arXiv 2404.02842, arXiv 2112.13128). Additionally, I will discuss a new proof of an inequality established by Matthieu Fradelizi, Mokshay Madiman, Mathieu Meyer, and Artem Zvavitch (arXiv 2206.02123).