Monotone and convex restrictions of continuous functions

Suppose that f belongs to a suitably defined complete metric space Cα of Hölder α-functions defined on [0,1]. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets A⊂[0,1] such that f|A is monotone, or convex/concave. Some of our results are about generic functions in Cα like the following one: we prove that for a generic f∈C1α[0,1], 0<α<2 for any A⊂[0,1] such that f|A is convex, or concave we have dimHA≤dim_MA≤max⁡{0,α−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<α≤2 for any f∈Cα[0,1] there is always a set A⊂[0,1] such that dimHA=α−1 and f|A is convex, or concave on A.