The Whitehead's theorem on the existence of convexity neighborhood on a Riemannian manifold has been a standard fact in Riemannian geometry. We give a curvature-free estimate on the size of convexity neighborhood around a point, which improve Whitedead's theorem and a recent result of Jiaqiang Mei, as well as related standard facts in Riemannian geometry. The estimate of injectivity radius has been very important in studying topology/geometry of manifold. Cheeger-Gromov-Taylor and Cheng-Li-Yau independently proved that the injectivity radius decays exponentially along a minimal geodesic, whose decay speed depends on the lower bound of Ricci curvature or sectional curvature. Jiaqiang Mei proved that the injectivity radius always admits a local Lipschitz decay under nonpositive sectional curvature. We prove that, in absence of conjugate points, the injectivity radius always admits a local Lipschitz decay, which improves all earlier known results. Both of our results are sharp in general, and was published online https://doi.org/10.1142/S0219199717500602.