Projection constants of symmetric spaces

We consider the problem of calculating the absolute projection constants $\lambda(X^n)$ of the symmetric n-dimensional spaces $X^n$. We find an integral representation of the projectors belonging to a class sufficient for calculation of the projection constants. The formula so obtained is used to calculate the absolute projection constant of one of the Marcinkiewicz spaces and, by so doing, to give a negative answer to a question raised by Grünbaum concerning the asymptotic behavior of the quantity $\lambda_n=\sup\limits_{X^n}\lambda(X^n)$.