A variant of Rudin's inequality for translation invariant projections on group algebras and its generalization to finite dimensional Banach spaces

Let G be a finite abelian group, let E be a MB° translation invariant subspace of the group algebra L₁ (G) and let P_E denote the translation invariant projection of L₁ (G) onto E. Rudin's averaging procedure yields the inequality ∥P_E∥L1(G) = 1/2∥∫_GT_g^-1(P + P^t)T _gdm∥ ≦1/2∥ P + P^t ∥_L1(G) for every projection P of L ₁ (G) onto E, where P_t denotes the projection on L₁ (G) = ℓ₁^N which is represented by the transpose of the matrix P. This inequality is extended to any N dimensional real Banach space X = (ℜ^N,∥·∥ ) as follows: An orthogonal projection P on X is called approximately minimal (a.m. in short) if there exist positive constants D and δ₀ so that, for every 0 < δ < δ₀, the ball B(P, δ) in the space L(X) of operators on X contains no orthogonal projection Q with ∥Q∥ < ∥P∥ (1 - Dδ²). Let P_E denote the orthogonal projection of X onto a subspace E. It is proved that P_E is a.m. if, and only if, ∥P_E∥ ≦1/2 ∥ P + P^t ∥ for every projection P of X onto E.