单形中Weitzenbck不等式和Sallee-Alexander不等式的稳定性

The stability of Weitzenbck inequality and Sallee-Alexander inequality for a simplex

对于n(n≥2)维Euclidean空间中n维单形的几何不等式,其径向函数或支撑函数很难找到,一般很难用径向或Hausdorff来度量2个单形的"偏差",使得对有关单形的几何不等式稳定性的研究比较困难.利用n维单形与其共超球的n维正则单形的偏差,引进了单形"R-偏正"度量的概念,证明了Gerber不等式、Euler不等式、SalleeAlexander不等式以及Weitzenbck不等式是稳定的,并给出这些几何不等式的稳定性版本.

It is very difficult to find the formula of radial function or support function of the simplex in n-dimensional Euclidean space En （n≥2）. Therefore, the deviation metric of the two simplices is difficult to be realized by radial metric or Hausdorff metric. The research on stability of geometric inequalities of simplices is also difficult. In this paper,by using the deviation of an n-simplex and a regular n-simplex which are on an （n-1）-dimensional hyper sphere, the r-deviation regular metric are introduced. Futher, by applying the R-deviation regular metric, we proved that Gerber inequality, Euler inequality, Sallee-Alexander inequality and Weitzenbock inequality with an n-simplex are all stable, and gave the stability versions to these geometric inequalities for a simplex.