Uniqueness of invariant Hahn-Banach extensions

Let ` be a linear functional on a subspace Y of a real linear space X provided with a sublinear functional p with ` p on Y . If G is an abelian semigroup of linear transformations T : X ! X such that T(Y ) Y , p(Tx) p(x) and `(Ty) = `(y) for all T 2 G, x 2 X and y 2 Y respectively, then a generalization of the classical Hahn-Banach theorem asserts that there exists an extension e` of `, e` p on X and e` remains invariant under G. The present paper investigates various equivalent conditions for the uniqueness of such extensions and these are related to nested sequences of p-balls, a concept that has proved useful in recent years in dealing with such extensions. The results are illustrated by a variety of examples and applications