A gap phenomenon on Riemannian manifolds with reverse volume pinching (Q2460644)

The main results of the article read as follows. (1) There exists a constant \(\eta = \eta (n) >0\) such that any closed simply connected manifold \(M\) of dimension \(n\) (\(n\geq 3\)), sectional curvature \(K_M\leq 1\), Ricci curvature \(\text{Ric}\,(M) \geq (n+2)/4\) and volume \(V(M) \leq (3/2)(1 + \eta)V(S^n)\), \(S^n\) being the unit sphere, has diameter \(d_M < 3\pi/2\). (2) There exist positive constants \(\delta = \delta (n)\) and \(\varepsilon = \varepsilon (n)\) such that any manifold as above with volume \(V(M) \leq (3/2)(1 + \delta)V(S^n)\) satisfies \(d_M\leq 3\pi/2 - \varepsilon\).