关于R^n空间上二次函数的反问题

The Inverse Problem on Property of Quadratic Function in n-Dimensional Euclidean Space

在研究了Rn空间上二次函数切平面的一个重要性的基础上,进一步研究其反问题,即证明了定义于Rn空间上的任意一个纯量函数f(x),如果它在点x1和x2处切平面的交线始终包含连接点x1和x2线段的中点,那么f(x)必为二次函数.

The author studied an important property of quadratic function on the tangent plane in n-dimensional Euclidean space. This result published on journal of Kashgar Teachers College in 2012,33（6）,and its title was“The property on Tangent Planes of a Quadratic Funcion in Rn”.In this paper,we further its inverse problem, that it is proved that a scalar-valued function f（x） defined in n-dimensional Euclidean space must be quadratic,if the intersection of tangent planes x1 and x2 always contains the midpoint of the line joining x1 and x2 .