A lower bound for the curvature integral under an upper curvature bound

It is proved that the integral of the scalar curvature over a Riemannian manifold is uniformly bounded below in terms of its dimension, upper bounds on the sectional curvature and volume, and a lower bound on the injectivity radius. This is an analog of an earlier result of Petrunin for Riemannian manifolds with sectional curvature bounded below.