×

zbMATH — the first resource for mathematics

Representation of functions of several variables by difference of convex functions. (English. Russian original) Zbl 0906.26006
J. Math. Sci., New York 100, No. 3, 2209-2227 (2000); translation from Zalgaller, V. A. (ed.) et al., Geometry and topology. 2. Work collection. Sankt-Peterburg: Matematicheskij Institut Im. V. A. Steklova, Sankt-Peterburgskoe Otdelenie, RAN, Zap. Nauchn. Semin. POMI. 246, 36-65 (1997).
Summary: If a function \(f: D^n\to\mathbb{R}\), where \(D^n\) is a convex compact set in \(\mathbb{R}^n\), admits a decomposition \(f= g-h\) with convex \(g\), \(h\), where \(h\) is upper bounded, then there exists such a decomposition which is in some sense “minimal”. A recurrent procedure converging to that decomposition is given. For piecewise linear functions \(f\), finite algorithms of those decompositions for \(n= 1,2\) are given. A number of examples clarifying some unexpected effects is presented and problems are formulated.

MSC:
26B40 Representation and superposition of functions
26B25 Convexity of real functions of several variables, generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. D. Aleksandrov, ”On surfaces representable by a difference of convex ones,” Izv. Akad. Nauk Kazakh. SSR. Ser. Mat. Mekh., 60, 3–20, (1949).
[2] A. D. Aleksandrov, ”Surfaces representable by a difference of convex ones,” Dokl. Akad. Nauk SSSR, 62, 613–616, (1950).
[3] I. Ya. Bakel’man, Geometric Methods of Solution for Elliptic Equations [in Russian], Moscow (1965).
[4] Yu. D. Burago and V. A. Zalgaller, ”Sufficient conditions of convexity,” Zap. Nauchn. Semin. LOMI, 45, 3–53, (1974).
[5] V. A. Zalgaller, ”Representation of a function of two variables by a difference of convex functions,” Vestn. Leningr. Univ., No. 1, 44–45, (1963).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.