This paper is to generalize convex functions to \(p\) -convex functions to dual \(p\) -convex functions. Concepts such as \(p\) -mean, Legendre transformation, convex base, and \(p\) -sum are described for dual \(p\) -convex functions. In particular, both the differential formula for dual \(p\) -convex functions and the reverse Borell–Brascamp–Lieb inequality are provided. Moreover, a reverse logarithmic Sobolev inequality is also proved by the dual Hamilton–Jacobi equations.
