Monotone and convex restrictions of continuous functions

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ofH\ddot{o}lder$ {\alpha}$-functionsdefinedon$[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets $A\subset [0,1]$ such that $f{\mathrm{\mid }}_A$ is monotone, or convex/concave. Some of our results are about generic functions in ${\mathcal{C}}^{\alpha }$ like the following one: we prove that for the generic $f\in C_1^{\alpha }[0,1]$, $0\le \alpha <2$ for any $A\subset [0,1]$ such that $f{\mathrm{\mid }}_A$ is convex, or concave we have ${$\mathrm{d}\mathrm{i}\mathrm{m}$}_{\mathrm H} A\leq \underline{$\mathrm{d}\mathrm{i}\mathrm{m}$}_M A\leq \max {0, {\alpha}-1 }.$ On the other hand, we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for $1<\alpha \le 2$ for any $f\in C^{\alpha }[0,1]$ there is always a set $A\subset [0,1]$ such that ${\mathrm{d}\mathrm{i}\mathrm{m}}_{\mathrm{H}}A=\alpha -1$ and $f{\mathrm{\mid }}_A$ is convex, or concave on $A$.