Hierarchy of deformations in concavity

A deformation is a positive continuous function defined on an appropriate interval. Through deformations, we generalize the notion of concavity for functions. We introduce the order function of a deformation, which permits to determine precisely the ranking of a deformation by taking account of the corresponding concavities. In the hierarchy, the action of positive constant multiples provides an equivalence relation and, if we focus on C^1-deformations, a one-to-one correspondence between the equivalence classes and the order functions is determined. Deformations having a constant valued order function play a fundamental role, and this is only the case of power functions. We show that the concavity associated to a deformation whose order function is nonincreasing and uniformly bounded from above by 𝑞∈ℝ can approximate the concavity associated to the power function of exponent q. Finally, we review three examples of deformations whose order function is nonincreasing and uniformly bounded from above. One is a power function, and the others are related to a concavity preserved by the Dirichlet heat flow in convex domains of Euclidean space.