Convex Analysis and Measurable Multifunctions
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580 C. Castaing M. Valadier
Convex Analysis and Measurable Multifunctions
Springer-Verlag Berlin· Heidelberg· New York 1977
Authors Charles Castaing Michel Valadier Universite des Sciences et Techniques du Languedoc Place Eugene Bataillon 34060 Montpellier CedexlFrance
Library of Cong res s Catalogi ng in Publica tion Dat a
Castaing, Charles, 1932-
Convex anaJ.ysis and measurab le multif\lllction s .
(Lecture notes in mathemat ic s ; 580 ) Includes bibliographies and ind ex.
l~ Functiona.l analysis . 2 . Convex fllnctions . Valadier, M. , 1940joint author . II. Titl e. III. Series : Lecture notes i n mathematics (Berlin) ; 580 QA3.I28 no . 580 [QA320) 510 '. 88 [515 ' .7] 77- 3987
1.
AMS Subject Classifications (1970) : 46XX ISBN 3-540-08144-5 Springer-Verlag Berlin ' Heidelberg· New York ISBN 0-387-08144-5 Springer-Verlag New York . Heidelberg · Berlin Th is work is subject t o copyright. All rig hts are reserved, whether the wh ole or part of the material is concern ed , specifically those of translatio n, reprintin g, re'use of illu strat ions, broad casting, reprodu ction by photocopying mac hine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, afee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1977 Printing and binding : Beltz Offsetdruck, Hemsbach f Bergstr. 2141f3140-543210
Preface
The present work is devoted to convex analysis, measurable multi-functions and some of their applications. The only necessary prerequisite for an intelligent reading is a good knowledge of analysis (Bourbaki or Dunford-Schwartz are appropriate references). of liftings of
Loo ;
One exception is the use
for their existence we refer to Ionescu-Tulcea 's
book. Nany questions are not treated, for example: the Borel selection theo rem due to Novikov, Arsenin, Kunugui ... ; the theory of set valued measures (Artstein, Coste, Drewnowsky, Godet-Thobie, Pallu de La Barriere ... ); the set valued martingales (Bismut, Daures, Neveu,
Van Cutsem ... ); the applicati on to optimal control and to the calculus of variations (Ekeland-Temam, Olech, Rockafellar ... ). Each chapter has its own bibli ography . Apologies are offered in advance to those who feel that they have been slighted. "Te take this opportunity to thank a small group of colleagues for their help in revising our manuscript. Finally, thanks are due to Mme Mori who typed most of the text, to H. Heyran and the whole secretari at of the department of mathematics.
Montpellier, October 1975
Contents
Chapter I. Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... . §
Convex lower semi-continuous functions. Bipolar theorem ....
2
§ 2
Some properties of convex sets ........ .......... ...... .....
7
§
3
Inf-compactness properties .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
§
4
Inf imum convolution • . . . . . . . . . . . . .
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