Links between functions and subdifferentials (Q1630600)

The author studies the links between functions and two notions of subdifferentials: the subdifferential determination and the subdifferential representation of functions. A function in a class $\mathcal{F}(X)$ is said to be subdifferentially determined in $\mathcal{F}(X)$ if it is equal up to an additive constant to any function in $\mathcal{F}(X)$ with the same subdifferential. A function is said to be subdifferentially representable if it can be expressed in terms of a subdifferential, or in other words, if it can be recovered from a subdifferential. The subdifferential determination and the subdifferential representation properties are studied with respect to an abstract subdifferential (covering the Clarke, the Michel-Penot and the Ioffe subdifferentials in any Banach space and others) and are studied in different classes of functions (like locally Lipschitz functions or extended-real-valued lower semicontinuous functions).