Riemannian convex bodies with a geodesic segment on the boundary

We construct compact convex bodies whose boundaries contain a nontrivial geodesic segment and are strictly convex elsewhere. The construction applies if sectional curvature is constant near a point, or if a normal Jacobi operator has a simple maximal eigenvalue. In particular, such a body exists in every smooth three-dimensional or four-dimensional Riemannian manifold.