凸体的两个几何量在Steiner对称化下的变化及其应用

Change of Two Geometric Quantities of Convex Bodies under Steiner Symmetrization and Application

本文研究了凸体的某些几何量在Steiner对称化作用下的变化情况.设C和D是d维欧氏空间R^d中的两个凸体,我们表明凸体关于特殊线性群SL(d)的极小表面积min{S(ΦC):Φ∈SL(d)}和混合体积V(C,D,…,D)分别在Steiner对称化作用后是不增加的.作为上述结果的应用,混合体积的Minkowski第一不等式被重新证明.

In this paper, we study some geometric quantities of convex bodies under Steiner symmetrization. Let C, D be two convex bodies in Euclidean d-space Rd. We show that the minimal surface area min{S（ΦC） ： Φ∈ SL（d）} with respect to the special linear group SL（d） and mixed volume V（C, D,…, D） of convex bodies are non-decreasing under Steiner symmetrization.As a consequence, the first Minkowski＇s inequality of mixed volume is proved again.