Convex bodies with minimal volume product in — a new proof

A new proof of the Mahler conjecture in R 2 is given. In order to prove the result, we introduce a new method — the vertex removal method; i.e., for any origin-symmetric polygon P , there exists a linear image ϕ P contained in the unit disk B 2 , and there exist three contiguous vertices of ϕ P lying on the boundary of B 2 . We can show that the volume-product of P decreases when we remove the middle vertex of the three vertices.