From Theorem 2.1 and 2.2 in Kanniappan (1983) "Uniform convergence of convex optimization problems", if a sequence of strictly convex functions $f_n$ uniformly converge to a strictly convex function $f$, then the sequence of solutions $x_n$ of $f_n$ converge to the unique solution $x_0$ of $f$.
Now, if you know that $||f_n-f||=o(g(n))$, is it true that $||x_n-x_0||=o(g(n))$?
And, if you know that $||f_n-f||=O(g(n))$ what could you say about $||x_n-x_0||$?
$O$ and $o$ are big and little O respectively.
Thank you in advance.