We extend Alberti's Rank-One Theorem to RCD(K, N ) metric measure spaces. Contents 1. Introduction 1.1. The Rank-One Theorem in the Euclidean setting 1.2. Main result 1.3. Outline of the proof Structure of the paper Acknowledgments 2. Preliminaries 2.1. Metric measure spaces 2.2. RCD spaces 3. Main results 3.1. Representation formula for the perimeter 3.2. Auxiliary results 3.3. Rank-One Theorem Appendix A. Rectifiability of the reduced boundary References

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We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties tr ⁡ ( Hess ⁡ f ) = Δ ⁢ f \operatorname{tr}(\operatorname{Hess}f)=\Delta f on U ⊆ X U\subseteq{\mathsf{X}} for every 𝑓 sufficiently regular, m = c ⁢ H n \mathfrak{m}=c\mathscr{H}^{n} on U ⊆ X U\subseteq{\mathsf{X}} for some c > 0 c>0 , where U ⊆ X U\subseteq{\mathsf{X}} is open and 𝖷 is a – possibly collapsed – RCD space of essential dimension 𝑛.

<p style='text-indent:20px;'>We study the stability of a class of action functionals induced by gradients of convex functions with respect to Mosco convergence, under mild assumptions on the underlying space.</p>

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