Upper Minkowski dimension estimates for convex restrictions (original) (raw)

We show that there are functions f in the Hölder class C α [0, 1], 1 < α < 2 such that f | A is not convex, nor concave for any A ⊂ [0, 1] with dim M A > α − 1. Our earlier result shows that for the typical/generic f ∈ C α 1 [0, 1], 0 ≤ α < 2 there is always a set A ⊂ [0, 1] such that f | A is convex and dim M A = 1. The analogous statement for monotone restrictions is the following: there are functions f in the Hölder class C α [0, 1], 1/2 ≤ α < 1 such that f | A is not monotone on A ⊂ [0, 1] with dim M A > α. This statement is not true for the range of parameters α < 1/2 and the main theorem of this paper for the parameter range 1 < α < 3/2 cannot be obtained by integration of the result about monotone restrictions.