Journal of Convex Analysis 13 (2006), No. 3, 623--632
Copyright Heldermann Verlag 2006
There are Many Totally Convex Functions
Dan Butnariu
Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel
dbutnaru@math.haifa.ac.il
Simeon Reich
Dept. of Mathematics, Technion - Israel Inst. Technology, 32000 Haifa, Israel
sreich@tx.technion.ac.il
Alexander J. Zaslavski
Dept. of Mathematics, Technion - Israel Inst. Technology, 32000 Haifa, Israel
ajzasl@tx.technion.ac.il
[Abstract-pdf]
Let $K$ be a convex subset of a normed linear space and let $R^1$ denote the real line. We show that there are many (in the sense of Baire category) strictly convex and totally convex functions $f \colon K \to R^1$. It is known that the existence of such functions is crucial in numerous optimization algorithms.
Keywords: Complete metric space, essentially strictly convex function, generic property, strictly convex function, totally convex function.
MSC: 46N10, 52A41, 54E50, 54E52
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