Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?

For the Fubini–Study metric on $\mathbb CP^n$ (it has positive 1/4pinched curvature). It has totally geodesic embedded $\mathbb CP^{n1}$ has codimension $2$. Then you can construct more larger codimension ones. And more trivial examples are given by closed geodesics in any positively curved space. Edit: If you prefer nonRiemannian Alexandrov space, take the spherical suspension of $\mathbb CP^n$. 


The obvious answer is an equatorial sphere (= intersection with a linear subspace of any codimension) in the unit sphere of $\mathbb{R}^n$. Without more details on your motivation, it is difficult to judge whether this answer is satisfying or not. 

