The set of balanced points with respect to 𝑆¹ and 𝑆³ actions of maps into Banach space

Abstract

Let G G be the group of units in the field F F , which is either R R , C C or H H , let X X be a free G G -space, and let f f be a map from X X to a Banach space E E over F F . In this paper we give an estimate for the size of the subset of X X consisting of points at which the average of f f is equal to zero. The result represents an extension of the Borsuk-Ulam-Yang theorem.