Convex analysis

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A branch of mathematics occupying a place intermediate between analysis and geometry, the subject of which are convex functions, convex functionals and convex sets (cf. Convex function (of a real variable); Convex functional; Convex set). The foundations of convex analysis were laid by H. Minkowski [1], [2], who created convex geometry, i.e. the geometry of convex sets in a finite-dimensional space. Many concepts of convex geometry found their culmination in functional analysis. The studies of W. Fenchel [3] opened a new stage in convex analysis, involving a detailed study of convex functionals. The formulation of convex analysis as a branch of mathematics in its own right took place in the 1950s and 1960s. The concepts and methods of convex analysis have found extensive application in various branches of mathematics: in the theory of extremal problems, especially so in convex programming and in the classical calculus of variations, in mathematical physics, in the theory of entire functions, in mathematical statistics, etc.

Fundamental concepts of convex analysis are polars, subdifferentials and conjugate functions (cf. Polar; Subdifferential; Conjugate function). The theorems of convex analysis relate the operations of conjugation, passing to a polar and taking a subdifferential with algebraic, set-theoretic and ordering operations over convex sets and functions. Other subjects of study include all possible dual relations between sets and their polars, functions and their conjugates, sets and homogeneous convex functions, etc.


[1] H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)
[2] H. Minkowski, "Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs" , Gesammelte Abhandlungen , 2 , Teubner (1911) pp. 131–229
[3] W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77
[4] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970)


A very important notion in convex analysis is that of duality.

For the modern development of convex analysis see the standard book [a1]. For the various applications of convex analysis and their close relations to other parts of geometry (as, e.g., boundary structure, geometric measure theory, optimization, etc.) see the excellent surveys and books [a2][a5].


[a1] J.T. Marti, "Konvexe Analysis" , Birkhäuser (1977)
[a2] R. Schneider, "Boundary structure and curvature of convex bodies" P.M. Gruber (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59
[a3] D. Gale, V. Klee, R.T. Rockafellar, "Convex functions on convex polytopes" Proc. Amer. Math. Soc. , 28 (1968) pp. 867–873
[a4] H.W. Kuhn, A.W. Tucker, "Nonlinear programming" , Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950) , Univ. California Press (1951) pp. 481–492
[a5] H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108
How to Cite This Entry:
Convex analysis. V.M. Tikhomirov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098