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Legendre Functions Whose Gradients Map Convex Sets to Convex Sets

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Computational and Analytical Mathematics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 50))

Abstract

We show that a differentiable function on a Banach space of dimension at least two has an affine gradient provided the gradient is continuous and one-to-one and maps convex sets to convex sets.

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Acknowledgements

We are grateful to an anonymous referee for several suggestions that helped us improve the presentation of this note.

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Authors and Affiliations

  1. Department of Mathematics, La Sierra University, Riverside, CA, 92515, USA

    Alexander Knecht & Jon Vanderwerff

Authors
  1. Alexander Knecht
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  2. Jon Vanderwerff
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Corresponding author

Correspondence to Jon Vanderwerff .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    David H. Bailey

  2. Mathematics, University of British Columbia, Kelowna, British Columbia, Canada

    Heinz H. Bauschke

  3. Dept. Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

    Peter Borwein

  4. Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

  5. Mathematics and Computer Science, University of Limoges, Limoges, France

    Michel Théra

  6. Mathematics and Computer Science Department, La Sierra University, Riverside, California, USA

    Jon D. Vanderwerff

  7. Dept. Combinatorics & Optimization, University of Waterloo Faculty of Mathematics, Waterloo, Ontario, Canada

    Henry Wolkowicz

Additional information

Dedicated to Jonathan Borwein on the occasion of his 60th birthday

Communicated By Heinz H. Bauschke.

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Knecht, A., Vanderwerff, J. (2013). Legendre Functions Whose Gradients Map Convex Sets to Convex Sets. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_21

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