从积分几何的观点看几何不等式——纪念李国平院士吴新谋教授诞辰100周年

该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式.

Geometric Inequalities-From Integral Geometry Point of View

This paper first surveys geometric inequalities achieved mainly by the Chinese mathematicians. By estimating the containment measure of a random convex body to be contained in, or to contain, another convex body via the fundamental kinematic formula of Blaschke and the formula of Poincarè in plane integral geometry, we obtain the classical isoperimetric inequality and some Bonnesen-style inequalities. Then some new geometric inequalities, such as the symmetric mixed isoperimetric inequality, Minkowski  and Bonnesen style symmetric mixed isohomothetic inequalities, are obtained. We also investigate the Gage type isoperimetric inequalities and the Ros type isoperimetric inequalities.