In this paper, we prove that if K is a convex body in $E^n$ and $E_i$ and $E_o$ are inscribed ellipsoid and circumscribed ellipsoid of K respectively with ${alpha}E_i=E_o$, then $[({alpha})^{frac{n}{p}+1}]^n{omega}^2_n{geq}V(K)V({Gamma}^{ast}_pK){geq}[(frac{1}{alpha})^{frac{n}{p}+1}]^n{omega}^2_n$. Lutwak and Zhang[6] proved that if K is a convex body, ${omega}^2_n=V(K)V({Gamma}_pK)$ if and only if K is an ellipsoid. Our inequality provides very elementary proof for their result and this in turn gives a lower bound of the volume product for the sets of constant width.