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Lipschitz continuity and interpolation properties of some classes of increasing functions. (English) Zbl 1034.26006

The author studies the classes of increasing positively homogeneous (IPH), increasing convex-along-rays (ICAR) and increasing co-radiant (ICR) functions on the positive orthant of \(\mathbb R^n.\) He proves that any finite IPH or ICR function is locally Lipschitz on the whole orthant and any ICAR function is locally Lipschitz on the interior of its domain. He also shows that such functions can be interpolated on arbitrary subsets by suitably defined min-type functions and gives two interesting characterizations of the IPHfunctions that satisfy the identity \(p\left( x_{1},...,x_{n}\right) =\left( p\left( x_{1}^{-1},...,x_{n}^{-1}\right) \right) ^{-1}\).

MSC:

26B35 Special properties of functions of several variables, Hölder conditions, etc.
26B25 Convexity of real functions of several variables, generalizations
26B05 Continuity and differentiation questions
90C26 Nonconvex programming, global optimization
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