Subspaces of codimension two with large projection constants

Let V be an n-dimensional real Banach space and let λ ( V ) denote its absolute projection constant. For any N ∈ N , N ≥ n define λ n N = sup ⁡ { λ ( V ) : dim ⁡ ( V ) = n , V ⊂ l ∞ ( N ) } . The aim of this paper is to determine minimal projections with respect to l 1 -norm as well as with respect to l ∞ -norm for subspaces given by solutions of certain extremal problems. As an application we show that for any n , N ∈ N , N ≥ n there exists an n-dimensional subspace V n ⊂ l 1 ( N ) such that λ n N = λ ( V n , l 1 ( N ) ) . Also we calculate relative and absolute projection constants of some subspaces of codimension two in l 1 ( N ) and l ∞ ( N ) for N ≥ 3 being odd natural number. Moreover, we show that for any odd natural number n ≥ 3 , λ n n + 1 < max x ∈ [ 0 , 1 ] ⁡ f n ( x ) ≤ λ n n + 2 , where f n ( x ) = 2 n n + 1 ( 1 − x ) + 1 2 ( x − 2 1 − x n + 1 + ( 2 1 − x n + 1 − x ) 2 + 4 ( 1 − x ) x ) . Also for any n ∈ N x n ∈ [ 0 , 1 ] satisfying f n ( x n ) = max x ∈ [ 0 , 1 ] ⁡ f n ( x ) will be calculated.