Approximate convexity and an edge-isoperimetric estimate

We study extremal properties of the functionF(x):=min{k‖x‖1−1/k:k⩾1},x∈[0,1], where ‖x‖=min{x,1−x}. In particular, we show that F is the pointwise largest function of the class of all real-valued functions f defined on the interval [0,1], and satisfying the relaxed convexity conditionf(λx1+(1−λ)x2)⩽λf(x1)+(1−λ)f(x2)+|x2−x1|,x1,x2,λ∈[0,1] and the boundary condition max{f(0),f(1)}⩽0. As an application, we prove that if A and S are subsets of a finite abelian group G, such that S is generating and all of its elements have order at most m, then the number of edges from A to its complement G∖A in the directed Cayley graph induced by S on G is∂S(A)⩾1m|G|F(|A|/|G|).