Lipschitz symmetric functions on Banach spaces with symmetric bases

We investigate Lipschitz symmetric functions on a Banach space X with a symmetric basis. We consider power symmetric polynomials on l(1) and show that they are Lipschitz on the unbounded subset consisting of vectors x is an element of l(1) such that vertical bar xn vertical bar <= 1. Using functions max and min and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space c(0) which can be described as a semiring of compositions of tropical polynomials over c(0).