Chieu, Nguyen Huy; Yao, Jen-Chih; Yen, Nguyen Dong Convexity of sets and functions via second-order subdifferentials. (English) Zbl 1460.49012 Linear Nonlinear Anal. 5, No. 2, 183-199 (2019). The aim of this article is to prove that the positive semidefiniteness of the limiting second-order subdifferential of the indicator function of a closed set with a \(C^2\)-smooth and regular boundary can characterize its local convexity. The authors also prove that such second-order characterization of local convexity is valid for any finite-dimensional closed set which locally can be represented as the epigraphs of piecewise \(C^2\) functions or the epigraphs of \(C^1\) functions. Further, they study the relationships between the convexity of a continuous function defined on a closed convex set and the positive semidefiniteness of its limiting second-order subdifferential with respect to the linear space generated by the set. Reviewer: Chandan Kumar Mondal (Durgapur) MSC: 49J52 Nonsmooth analysis 26B25 Convexity of real functions of several variables, generalizations Keywords:second-order subdifferential; closed set; indicator function; smooth boundary; local convexity; continuous function; convexity PDFBibTeX XMLCite \textit{N. H. Chieu} et al., Linear Nonlinear Anal. 5, No. 2, 183--199 (2019; Zbl 1460.49012) Full Text: Link