Approximate convexity and an edge-isoperimetric estimate

We study extremal properties of the function F ( 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 condition f ( λ x 1 + ( 1 − λ ) x 2 ) ⩽ λ f ( x 1 ) + ( 1 − λ ) f ( x 2 ) + | x 2 − x 1 | , x 1 , x 2 , 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 ) ⩾ 1 m | G | F ( | A | / | G | ) .