The Mahler Conjecture in Three Dimensions

Abstract

The Mahler conjecture dates back to 1938. This paper solves the conjecture for general convex bodies in three dimensions by developing a method called the shadow flow. The equality case is characterized as well. This method is also applied to give a new proof of the three-dimensional symmetric case, which was first proved by Iriyeh--Shibata.

This version combines two earlier papers into one: the paper on the non-symmetric Mahler conjecture for general convex bodies and the symmetric Mahler inequality paper (arXiv:2605.13795). Full equality characterizations are included. The symmetric paper (arXiv:2605.13795) will not be submitted separately and is unlikely to be updated further