A noncontinuous generalization of the arithmetic–geometric mean

摘要

The notions of prickly set, scalar and vectorial mean are defined. A noncontinuous generalization of the arithmetic–geometric mean is given, by considering the recursion xn+1=F(xn), where F:C→C is a vectorial mean and C is a closed prickly subset of Rm. The convergence of this recursion is proved and it is shown that the limit is contained in the diagonal of C. If F is continuous, it is deduced that the limit of the recursion is a continuous function of the initial value x=x0. Denoting the limit by F∞(x) it is proved that if F is monotone, then F∞ it is also monotone (where the monotonicity is considered with respect to the closed cone R+m).