We prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent n (n \ge 2 ), then it has exactly the  n-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if a complete  n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Gagliardo-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. The other one is that we give an alternative proof to the main result in [J. Mao, The Gagliardo-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature, to appear in Kyushu J. Math.] for smooth metric measure spaces with nonnegative weighted Ricci curvature. This is a joint-work with Feng Du, Qiao-Ling Wang and Chuan-Xi Wu.