I'm trying to understand which Riemannian manifolds can be Lipschitz approximated by polyhedral spaces of the same dimension with curvature bounded below. The necessary conditions I found consist of some special inequality for curvature at each point (the geometric curvature bound). This inequality is also sufficient condition for local approximation. I conjecture that it is also a sufficient condition for global approximation, and I can prove it if the curvature bound is positive. In general I can prove it only with the additional assumption that tangent bundle of the manifold is stably trivial.
PETRUNIN, ANTON (2003) "Polyhedral approximations of Riemannian manifolds," Turkish Journal of Mathematics: Vol. 27: No. 1, Article 9. Available at: https://journals.tubitak.gov.tr/math/vol27/iss1/9