Takagi functions and approximate midconvexity

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