  The purpose of this paper is to understand the geometric structure of the
$L^2$-Wasserstein space $\pp$ over the Euclidean space.For this sake, we focus
on its cone structure.One of our main results is that the $L^2$-Wasserstein
space over a Polish space has a cone structure if and only if so does the
underlying space.In particular, $\pp$ turns out to have a cone structure.It is
also shown that $\pp$ splits $\R^d$ isometrically but not $\R^{d+1}$.
