  We prove that if a metric measure space satisfies the volume doubling
condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent
$n \ge 3$, then it has exactly the $n$-dimensional volume growth. As an
application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci
curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp
constant, then its flag curvature is identically zero. In the particular case
of Berwald spaces, such a space is necessarily isometric to a Minkowski space.
