Convex bodies with minimal volume product in R2R2 — a new proof

A new proof of the Mahler conjecture in R2R2 is given. In order to prove the result, we introduce a new method — the vertex removal method  ; i.e., for any origin-symmetric polygon PP, there exists a linear image ϕPϕP contained in the unit disk B2B2, and there exist three contiguous vertices of ϕPϕP lying on the boundary of B2B2. We can show that the volume-product of PP decreases when we remove the middle vertex of the three vertices.