Abstract
The main result is an equality type mean value theorem for tangentially convex functions in terms of tangential subdifferentials, which generalizes the classical one for differentiable functions, as well as Wegge theorem for convex functions. The new mean value theorem is then applied, analogously to what is done in the classical case, to characterize, in the tangentially convex context, Lipschitz functions, increasingness with respect to the ordering induced by a closed convex cone, convexity, and quasiconvexity.
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Acknowledgements
I gratefully acknowledge financial support from the Spanish Ministry of Science, Innovation and Universities, through the grant PGC2018-097960-B-C21 and the Severo Ochoa Program for Centers of Excellence in R&D (CEX2019-000915-S). I am affiliated with MOVE (Markets, Organizations and Votes in Economics). I am greatly indebted to Soghra Nobakhtian for having posed me the question whether tangentially convex functions admit a mean value theorem, to two anonymous reviewers for helpful remarks, and to Jean-Baptiste Hiriart-Urruty for having provided me with a scanned copy of [13] (a very interesting paper, which, unfortunately, appears not to be available online).
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Dedicated to Miguel Ángel Goberna on the occasion of his 70th birthday
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Martínez-Legaz, J.E. A Mean Value Theorem for Tangentially Convex Functions. Set-Valued Var. Anal 31, 13 (2023). https://doi.org/10.1007/s11228-023-00674-3
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DOI: https://doi.org/10.1007/s11228-023-00674-3
Keywords
- Mean value theorem
- Tangential convexity
- Tangential subdifferential
- Convexity
- Monotonicity
- Quasiconvexity
- Quasimonotonicity