The aim of this talk is to present some results about the notions of convexity for a hypersurface in a Finsler manifold. The main theorem concerns the infinitesimal and local notions of convexity which are shown to be equivalent. Using a different approach, this result extends the classical Bishop\\\\\\\'s one for the Riemannian case to the Finsler setting. 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.