Takagi functions and approximate midconvexity

Let V   be a convex subset of a normed space and let ε⩾0ε⩾0, p>0p>0 be given constants. A function f:V→Rf:V→R is called (ε,p)(ε,p)-midconvex iff(x+y2)⩽f(x)+f(y)2+ε‖x−y‖pfor all x,y∈V. We consider the case p∈[1,2]p∈[1,2] and investigate the relations between continuous (ε,p)(ε,p)-midconvex functions and Takagi-like functions given byωp(x):=∑k=0∞22kpdist(2kx;Z)for x∈R. It occurs that functions ωpωp are optimal (1,p)(1,p)-midconvex functions. This gives us sharp estimations for continuous (ε,p)(ε,p)-midconvex functions.We also compute the maximum of the function ωpωp for a certain set of parameter values.