Equipped with the
L
2
,
q
L^{2,q}
-distortion distance
\DD _{2,q}, the space
\XX _{2q} of all metric measure spaces
(X,\d ,\m ) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on
\ol \XX _{2q} are presented.