Origin-Symmetric Bodies of Revolution with Minimal Mahler Volume in   R^3-a new proof
  Meyer and Reisner had proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler volume. 
Youjiang Lin and Gangsong Leng
math.DG