John and Lowner ellipsoid

Question

I am looking at a proof to show that Lowner ellipsoids are unique for centrally symmetric convex body $K$. I want to show basically that $$ \DeclareMathOperator{\Vol}{Vol}\DeclareMathOperator{\Low}{Low}\DeclareMathOperator{\John}{John} \Low(K)=\John(K^{\circ})^{\circ}, $$ where the $\circ$ means polar sets. To do so, I want to show that the minimization problem to find the Lowner ellipsoid is the same as the maximization problem to find the John ellipsoid on the polar convex body $K^{\circ}$. The proof says that it is trivial since $$ \Vol\big(\Low(K)\big)\cdot \Vol\big(\Low(K)^{\circ}\big)$$ equals the volume of the unit ball squared, and that $$ K \subset \Low(K) \iff \Low(K)^\circ \subset K. $$ I don't know how to connect the pieces together, however.