Institution: Stanford University, Department of Computer Science

Title: The maximum and minimum of a positive definite quadratic polynomial on a sphere are convex functions of the radius

Author: Forsythe, George E.

Date: July 1969

Abstract: It is proved that in euclidean n-space the maximum M($\rho$) and minimum m($\rho$) of a fixed positive definite quadratic polynomial Q on spheres with fixed center are both convex functions of the radius $\rho$ of the sphere. In the proof, which uses elementary calculus and a result of Forsythe and Golub, $m^" (\rho) and M^" (\rho)$ are shown to exist and lie in the interval [$2{\lambda}_1 ,2{\lambda}_n$], where ${\lambda}_i$ are the eigenvalues of the quadratic form of Q. Hence $m^" (\rho) > 0 and M^" (\rho) > 0$.

http://i.stanford.edu/pub/cstr/reports/cs/tr/69/144/CS-TR-69-144.pdf