Polynomials as Lipschitz maps on the Veronese cone

Given a Banach space $X$ and $d\in \mathbb{N}$, we construct a metric space $\mathbb{V}_X^d$ with the property that every $d$-homogeneous polynomial defined on $X$ factors through a Lipschitz map on it. We prove that the metric on $\mathbb{V}_X^d$ is independent (up to a constant) of the norm of the tensor space in which it is embedded. We apply this fact to prove that a homogeneous polynomial is Lipschitz $q$-summing as a polynomial if and only if its associated Lipschitz map is Lipschitz $q$-summing. This result generalizes the already known theorem for linear operators