Geodesics on non-complete Finsler manifolds

In this note based on paper BARTOLO R, CAPONIO E, GERMINARIO A, SÀNCHEZ M (2011). Convex domains of Finsler and Riemannian manifolds. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, we deal with domains D (i.e. connected open subsets) of a Finsler manifold (M,F). At first we carry out a comparison between different notions of convexity for their boundaries. Then a careful application of variational methods to the geodesic problem yields that the convexity of the boundary of D is equivalent to the existence of a minimal geodesic for each pair of points of D. Furthermore multiplicity of connecting geodesics can be obtained if D is not contractible.