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Topical and sub-topical functions, downward sets and abstract convexity. (English) Zbl 1007.26010

A function \(f:\mathbb R^{n}\to\overline{\mathbb R}\) is called topical if it is increasing (in the natural partial ordering \(\leq\) of \(\mathbb R^{n})\) and satisfies the equality \(f( x+\lambda\mathbf{1}) =f( x) +\lambda\) for every \(x\in\mathbb R^{n}\) and \(\lambda\in\mathbb R\) (\(\mathbf{1}\in\mathbb R^{n}\) denotes the vector with all coordinates equal to \(1)\); an increasing function \(f\) satisfying the inequality \(f( x+\lambda\mathbf{1}) \leq f(x) +\lambda\) is called sub-topical. A subset \(G\) of \(\mathbb R^{n}\) is said to be downward if \(g\in G\) implies that \(x\in G\) for every \(x\leq g.\) This paper studies the relationship between these notions; for instance, it proves that the plus-Minkowski gauge \(\rho_{G}:\mathbb R^{n}\to\overline{\mathbb R}\) of a downward set \(G\) (i.e., the function defined by \(\rho_{G}( x) =\inf\{ \lambda\in\mathbb R: x\in\lambda\mathbf{1}+G\})\) is topical. It also develops an abstract convexity framework for studying them, based on the coupling function \(\varphi:\mathbb R^{n}\times\mathbb R^{n}\to\mathbb R\) defined by \(\varphi( x,w) =\min_{1\leq i\leq n}\{ x_{i}+w_{i}\}\) \(( x=( x_{i}) ,w=( w_{i}) \in\mathbb R^{n}) ;\) in particular, it characterizes topical functions as those functions \(f\) whose Fenchel-Moreau conjugates with respect to \(\varphi,\)\(f^{c( \varphi) },\) satisfy \(f^{c( \varphi) }(w) =-f( -w)\) \(( w\in\mathbb R^{n}) ,\) and shows that the largest topical minorant of any function \(f\) coincides with its Fenchel-Moreau biconjugate with respect to \(\varphi.\)

MSC:

26B25 Convexity of real functions of several variables, generalizations
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
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[1] Gaubert, S. and Gunawardena, J. 1998. A non-linear hierarchy for discrete event dynamical systems. Proc. 4th Workshop on discrete event systems. 1998, Cagliari. Technical Report HPL-BRIMS-98-20, Hewlett-Packard Labs,1998 · Zbl 0933.49017
[2] Gondran M., Comptes Rendus Acad. Sci 323 pp 371– (1996)
[3] DOI: 10.1017/CBO9780511662508.003 · doi:10.1017/CBO9780511662508.003
[4] Gunawardena J., Theoretical Computer Science (1999)
[5] Gunawardena J., Technical Report HPL-BRIMS-95-003 (1995)
[6] DOI: 10.1080/02331939408843993 · Zbl 0815.49026 · doi:10.1080/02331939408843993
[7] DOI: 10.1023/A:1009710513565 · Zbl 0899.49016 · doi:10.1023/A:1009710513565
[8] Pallaschke D., Foundations of Mathematical Optimization. Convex Analysis without Linearity (1997) · Zbl 0887.49001
[9] Penot J.P., Acta Math Vietnamica 22 pp 541– (1997)
[10] Rubinov A.M., Abstract Convexity and Global Optimization (2000) · Zbl 0985.90074 · doi:10.1007/978-1-4757-3200-9
[11] Rubinov A.M., Abstract Convex Analysis (1997)
[12] Singer I., Abstract Convex Analysis (1997) · Zbl 0898.49001
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