A minimal set-valued strong derivative for vector-valued Lipschitz functions |
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Authors: | T. H. Sweetser III |
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Affiliation: | (1) Department of Mathematics, University of California at San Diego, La Jolla, California |
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Abstract: | A set-valued derivative for a function at a point is a set of linear transformations whichapproximates the function near the point. This is stated precisely, and it is shown that, in general, there is not a unique minimal set-valued derivative for functions in the family of closed convex sets of linear transformations. For Lipschitz functions, a construction is given for a specific set-valued derivative, which reduces to the usual derivative when the function is strongly differentiable, and which is shown to be the unique minimal set-valued derivative within a certain subfamily of the family of closed convex sets of linear transformations. It is shown that this constructed set may be larger than Clarke's and Pourciau's set-valued derivatives, but that no irregularity is introduced.The author would like to thank Professor H. Halkin for numerous discussions of the material contained here. |
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Keywords: | Derivative set-valued derivative generalized derivative Lipschitz functions nondifferentiable functions convex sets linear transformations |
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