# Do Alexandrov spaces with non-empty boundary satisfy $RCD^*$ condition?

Let $M$ be an $n-$ dim compact Alexandrov space with curvature $\geq k$ with non-empty boundary $\partial M$.

Recently, a notion of generalized lower Ricci curvature bound on metric measure spaces has been defined, called $RCD^*$ condition.

Question: Does $M$ satisfy $RCD^*((n-1)k,n)$ condition ?

Remark: If M has no boundary, i.e., $\partial M=\varnothing$， then it is well known that $M$ satisfies $RCD^*((n-1)k,n)$ condition. The definition of $RCD^*$ condition is given in [1,2].

Reference:

[1] Erbar, M., Kuwada, K., Sturm, K.: On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces. Invent. Math. 201, 993–1071 (2015)

[2] Ambrosio, L., Mondino, A., Savaré, G.: On the Bakry–Émery condition, the gradient estimates and the local-to global property of RCD ∗ (K, N) metric measure spaces. J. Geom. Anal. 26(1), 24–56 (2016)