The Symplectic Mahler Conjecture for Balanced Convex Domains and Hanner Polytopes

A balanced convex domain in 𝑅 𝑛 is a compact convex set that is invariant under the (𝑍/2) 𝑛 action which flips the signs of the coordinates. In this talk, I will explain a construction of a simple symplectic embedding of a ball of capacity 4 into the product of 𝐾 and its polar body for every balanced convex domain 𝐾. If 𝐾 is a Hanner polytope, we also prove that this embedding is surjective.