We compare the notion of higher-dimensional convexity, as defined by Carriere, for real projective manifolds with the existence of hemispheres. We show that if an i-convex real projective manifold M of dimension n for an integer i with 0 < i < n has an i-dimensional hemisphere, then M is projectively homeomorphic to S-n/T where T is a finite subgroup of O(n + 1, R) acting freely on S-n.