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On Conjugate Convex Functions

Published online by Cambridge University Press:  20 November 2018

W. Fenchel
Affiliation:
The Technical University of Denmark Copenhagen
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Extract

Since the classical work of Minkowski and Jensen it is well known that many of the inequalities used in analysis may be considered as consequences of the convexity of certain functions. In several of these inequalities pairs of “conjugate” functions occur, for instance pairs of powers with exponents a and a related by 1/a + 1/a = 1. A more general example is the pair of positively homogeneous convex functions denned by Minkowski and known as the distance (or gauge) function and the function of support of a convex body. The purpose of the present paper is to explain the general (by the way rather elementary) idea underlying this correspondence.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Bonnesen, T., Fenchel, W., Théorie der konvexen Körper (Berlin, 1934).Google Scholar
[2] Hardy, G. H., Littlewood, J. E., Pólya, G., Inequalities (Cambridge, 1934).Google Scholar
[3] Mandelbrojt, S., “Sur les fonctions convexes,” C.R. Acad. Sci. Paris, vol. 209 (1939), 977978.Google Scholar

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