Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACT

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.

KEYWORDS:

• Semicontinuous functions
• upper and lower envelopes
• convex majorants
• positively homogeneous functions
• subdifferential

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49J52
• 49K27
• 26B40

Disclosure statement

There are no conflicts of interest to disclose.

ORCID

Valentin V. Gorokhovik  http://orcid.org/0000-0003-2447-5943

Additional information

Funding

The research was supported by the State Research Program of the Republic of Belarus [grant number ‘Convergence – 1.04.01’].