  Let $Y$ be a compact K\"ahler normal space and $\alpha \in
H^{1,1}(Y,\mathbb{R})$ a K\"ahler class. We study metric properties of the
space $\mathcal{H}_\alpha$ of K\"ahler metrics in $\alpha$ using Mabuchi
geodesics. We extend several results by Calabi, Chen, Darvas previously
established when the underlying space is smooth. As an application we
analytically characterize the existence of K\"ahler-Einstein metrics on
$\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate
these concepts in the case of toric varieties.
