New sharp affine isoperimetric inequalities for the volume decomposition functionals are established. To attack these extremal problems, we find the recursion formulas of volume decomposition functionals, then introduce concentration polytope and characterize its geometric structure, especially its vertices and 1-dimensional faces. The concentration polytope intuitively reinterprets the discrete logarithmic Minkowski problem.

熊革，同济大学教授，博士生导师。研究领域是凸体几何。

熊革教授解决了凸体几何中的几个公开问题。包括Lutwak-Yang-Zhang关于锥体积泛函极值问题的2, 3维情形；由截面确定凸体的Baker-Larman问题的2维情形。他与学生最早提出、并解决了Lp静电容量的Minkowski 问题；完全解决了纽约大学G. Zhang教授关于凸体的John 椭球与对偶惯性椭球一致性的问题。

熊革教授在国际纯数学的重要期刊JDG, AIM, IUMJ, IMRN, CVPDE, JFA，CAG, Israel Journal of Mathematics, Discrete and Computational Geometry等上发表论文30余篇。他的多个研究成果被写入凸体几何的经典教材《Geometric Tomography》和《Convex Bodies: the Brunn-Minkowski theory》之中。