Some Relations between Volume, Injectivity Radius, and Convexity Radius in Riemannian Manifolds

Abstract

One knows how to attach to a compact riemannian manifold (M, g) three real numbers: its volume υ(g), its injectivity radius i(g) and its convexity radius c(g). The present article studies the following problems: do there exist universal constants λ(n), µ(n) such that υ(g)≥λ(n)i ⁿ(g), υ(g)≥µ(n)c ⁿ(g) for every riemannian manifold of dimension n? An affirmative answer is given to the second problem for any n but with an unsharp constant; an affirmative answer is given to the first problem only when n = 2 but with the sharp bound λ(2) = 4/π.