Let M be a finite-dimensional C3 manifold supplied with a C2 Finsler metric ds = F(x, dx), which is not necessarily even in dx. Let p designate the induced oriented topological metric. For any p E M, the antipodal locus of p is the set A(p)= {qEMIp(p, q)_p(p, r) for all rEM}. For example, if M is a real projective space with the Riemannian metric of constant curvature 1, A (p) is a smooth hypersurface (in fact, a projective hyperplane) for every pEHM. One may ask, how close does this property come to characterizing real projective spaces among Finsler manifolds? We prove