Starting from the axiomatic field-theory principles (analyticity, unitarity and polynomial boundedness) we obtain high-energy bounds on the amplitude of elastic scattering in a space-time of general dimension D . This is of interest due to recently developed theories in higher dimensions. A comparison of our bounds (which are rigorously valid only for a theory with a mass gap) with the high-energy behaviour of string amplitude, summed up to all loops by Amati, Ciafaloni and Veneziano, shows that σ inel saturates our bound, while consistency with the behaviour of σ tot requires D ⩾ 6 and a fast shrinking of the analyticity ellipse (which might indicate nonlocality of the string theory). This behaviour of σ tot is due to massless graviton exchange and violates our bounds if no shrinking of the analyticity ellipse is assumed. Our fixed-angle bound, however, is obeyed by the high-energy fixed-angle behaviour of elastic string scattering amplitudes in all orders obtained by Mende and Ooguri. We also show that analyticity of the amplitude in s leads to the well-known extension of the analyticity domain. Alternatives to polynomial boundedness (relevant to string theories) are also discussed.