Neighborhoods on the Grassmannian of marginals with bounded isotropic constant

We show that for any isotropic log-concave probability measure μ   on RnRn, for every ε>0ε>0, every 1⩽k⩽n and any E∈Gn,kE∈Gn,k there exists F∈Gn,kF∈Gn,k with d(E,F)<εd(E,F)<ε and LπFμ