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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.

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