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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
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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).
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