Abstract

Due to recent results by the authors, Holditch's original proof of his eponymous theorem is effective when applied to a sufficiently smooth convex curve with positive curvature, since such curves were shown to possess a smooth envelope generated by a traveling chord. In the present work, we smoothly approximate a convex curve satisfying only a mild local obtuseness condition, and thereby derive Holditch's theorem for such curves without requiring them to possess an envelope.

Author information

Contact details are reproduced from the original publication and may be historical.

Ronald J. Stern

Dept. of Mathematics and Statistics, Concordia University, Montreal, Canada

ron.stern@concordia.ca

Keywords

  • Holditch's theorem
  • convex analysis
  • Broman's condition
  • local obtuseness
  • smooth approximation

Mathematics Subject Classification

26A27, 49J52

References

  1. T. M. Apostol, A. Mamikon, A. Mnatsakanian: Subtangents – an aid to visual calculus, Amer. Math. Monthly 109 (2002) 525–533.https://doi.org/10.1080/00029890.2002.11919882Google Scholar
  2. D. Azagra, D. Stolyarov: Inner and outer smooth approximation of convex hypersurfaces. When is it possible?, Nonlinear Analysis 230 (2023), art. no. 113225.https://doi.org/10.1016/j.na.2023.113225Google Scholar
  3. A. Broman: Holditch’s Theorem (A fresh look at a long-forgotten theorem), Math. Magazine 54/3 (1981) 99–108.https://doi.org/10.1080/0025570X.1981.11976908Google Scholar
  4. F. H. Clarke, Yu. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  5. Reverend H. Holditch: Geometrical theorem, Quart. J. 2 (1858) p. 38.Google Scholar
  6. H. Proppe, A. Stancu, R. J. Stern: On Holditch’s theorem and Holditch curves, J. Convex Analysis 24/1 (2017) 239–259.https://doi.org/10.68381/jca24017Google Scholar
  7. H. Proppe, A. Stancu, R. J. Stern: Holditch’s envelope, Comm. Optim. Theory 2022 (2022) 1–11.https://doi.org/10.23952/cot.2022.13Google Scholar
  8. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  9. R. Schneider: Convex Bodies and the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications Vol. 151, Cambridge University Press, Cambridge (2014).Google Scholar

Suggested citation

H. Proppe, A. Stancu, R. J. Stern. “Approximation Approach to Holditch's Theorem.” Journal of Convex Analysis 32 (2025), No. 3, 723–730. https://doi.org/10.68381/jca32035

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.