CD meets CAT

We show that if a noncollapsed $CD(K,n)$ space $X$ with $n\ge 2$ has curvature bounded above by $\kappa$ in the sense of Alexandrov then $K\le (n-1)\kappa$and$X$isanAlexandrovspaceofcurvatureboundedbelowby$K-\kappa (n-2)$\mathrm{.}Wealsoshowthatifa$CD(K,n)$space$Y$withfinite$n$ has curvature bounded above then it is infinitesimally Hilbertian.