  A detailed study of the notions of convexity for a hypersurface in a Finsler
manifold is carried out. In particular, the infinitesimal and local notions of
convexity are shown to be equivalent. Our approach differs from Bishop's one in
his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the
Riemannian case. Ours not only can be extended to the Finsler setting but it
also reduces the typical requirements of differentiability for the metric and
it yields consequences on the multiplicity of connecting geodesics in the
convex domain defined by the hypersurface.
