Rigidity of cones with bounded Ricci curvature

We show that the only metric measure space with the structure of an $N$-cone and with two-sided synthetic Ricci bounds is the Euclidean space ${\mathbb R}^{N+1}$for$N$ integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymtotic of the heat kernel in the $L^2$-transport distance. Moreover, we establish a beautiful rigidity results of independent interest which characterize the $N$-dimensional standard sphere ${\mathbb{S}}^N$ as the unique minimizer of $${\int }_X{\int }_X\cos d(x,y)m(dy)m(dx)$$ among all metric measure spaces with dimension bounded above by $N$ and Ricci curvature bounded below by $N-1$.