Improvement of the dimension bound for unweighted RCD spaces

Abstract

We show that if $(X,d,\mathscr H^n)$ is an $RCD(K,N)$ space for some $K\in\mathbb{R}$ and $N\in [1,\infty)$, then it is also a (non-collapsed) $RCD(K,n)$ space. In other words, unweighted finite-dimensional $RCD$ spaces enjoy an automatic improvement of the dimension bound to the Hausdorff dimension of the space. More generally, we prove that this holds also when $N=\infty$, provided that we assume in addition the $n$-rectifiability of the space.