The aim of this paper is to provide new characterizations of the
curvature dimension condition in the context of metric measure
spaces $(X,\mathsf d,\mathfrak m)$. On the geometric side, the
authors' new approach takes into account suitable weighted action
functionals which provide the natural modulus of $K$-convexity when
one investigates the convexity properties of $N$-dimensional
entropies. On the side of diffusion semigroups and evolution
variational inequalities, the authors' new approach uses the
nonlinear diffusion semigroup induced by the $N$-dimensional
entropy, in place of the heat flow. Under suitable assumptions
(most notably the quadraticity of Cheeger's energy relative to the
metric measure structure) both approaches are shown to be
equivalent to the strong $\mathrm {CD}^{*}(K,N)$ condition of
Bacher-Sturm.
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