I want to minimize the least squares objective
$$\underset{\boldsymbol p \in \mathbb{R}^{n_p}}{\text{min}} \lVert \boldsymbol f(\boldsymbol p) \rVert$$
where ${n_p}$ is the number of parameters on which the $f_i$ depend.
I am not looking for a strict mathematical proof but more for a practical guideline how to check if the objective is convex or not.
Given that I know the ranges of all, say ${n_p} = 4$, paramters, my idea was to plot $\lVert \boldsymbol f(p_i) \rVert$ for all $i=1,\dots,4$. If the 4 resulting curves are convex each, then the objective above is convex.
Is that true?