  The purpose of this article is to study the strict convexity of the Mabuchi
functional along a $C^{1,1}$-geodesic, with the aid of the
$\epsilon$-geodesics. We proved the $L^2$-convergence of the fiberwise volume
element of the $\epsilon$-geodesic. Moreover, the geodesic is proved to be
uniformly fiberwise non-degenerate if the Mabuchi functional is
$\epsilon$-affine.
