Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Convexity and constructive infima

Abstract

We show constructively that every quasi-convex uniformly continuous function \(f : \mathrm {C}\rightarrow \mathbb {R}^+\) has positive infimum, where \(\mathrm {C}\) is a convex compact subset of \(\mathbb {R}^n\). This implies a constructive separation theorem for convex sets.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Berger, J., Bridges, D.: The fan theorem and positive-valued uniformly continuous functions on compact intervals. N. Z. J. Math. 38, 129–135 (2008)

  2. 2.

    Berger, J., Svindland, G.: A separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov’s principle. Ann. Pure Appl. Log. 167, 1161–1170 (2016)

  3. 3.

    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)

  4. 4.

    Bishop, E., Bridges, D.: Constructive Analysis. Springer, Heidelberg (1985)

  5. 5.

    Bridges, D., Vîtă, L.S.: Techniques of Constructive Analysis, Universitext. Springer, New York (2006)

  6. 6.

    Ishihara, H.: Reverse mathematics in Bishop’s constructive mathematics. Philos. Sci Cah. Spec. 6, 43–59 (2006)

  7. 7.

    Julian, W., Richman, F.: A uniformly continuous function on \(\left[0,1\right]\) that is everywhere different from its infimum. Pac. J. Math. 111(2), 333–340 (1984)

Download references

Acknowledgments

We thank the referee and Christian Ittner for helpful comments.

Author information

Correspondence to Josef Berger.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Berger, J., Svindland, G. Convexity and constructive infima. Arch. Math. Logic 55, 873–881 (2016). https://doi.org/10.1007/s00153-016-0502-y

Download citation

Keywords

  • Bishop’s constructive mathematics
  • Brouwer’s fan theorem
  • Convex functions
  • Separating hyperplanes

Mathematics Subject Classification

  • 03F60
  • 52A41