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The Second‐Order Subdifferential and the Dupin Indicatrices of a Non‐Differentiable Convex Function
Author(s) -
Hiriart-Urruty J.-B.,
Seeger A.
Publication year - 1989
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-58.2.351
Subject(s) - mathematics , subderivative , pseudoconvex function , differentiable function , order (exchange) , concave function , regular polygon , pure mathematics , convex function , function (biology) , mathematical analysis , geometry , convex optimization , evolutionary biology , economics , biology , finance
We introduce the notions of Dupin indicatrices for ‘non‐smooth’ convex surfaces in R n +l which are graphs of convex functions defined on R n . We study them in connection with the concept of second‐order subdifferentials of convex functions, such as have been introduced and developed recently by the authors. Finally, second‐order subdifferentials are viewed as limit sets of difference quotients involving approximate subdifferentials, which sheds a new light on the ‘second‐order information’ contained in these approximate subdifferentials.