Collapsing vs. positive pinching

Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant i_o = i_o(M, δ) > 0 such that the injectivity radius of any Riemannian metric g on M with δ ≤ K_g ≤ 1 can be estimated from below by i_o. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the Klingenberg-Sakai conjecture: Given any metric d_o on M, there exists a constant i_o = i_o(M, d_o, δ) > 0, such that the injectivity radius of any δ-pinched d_o-bounded Riemannian metric g on M (i.e., dist_g ≤ d_o and δ ≤ K_g ≤ 1) can be estimated from below by i_o. We also establish a continuous version of the Klingenberg-Sakai conjecture, saying that a continuous family of metrics on M with positively uniformly pinched curvature cannot converge to a metric space of strictly lower dimension.