Harald Proppe, Alina Stancu, Ronald J. Stern, Holditch’s Envelope, Vol. 2022 (2022), Article ID 13, pp. 1-11

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DOI: 10.23952/cot.2022.13

Received April 14, 2022; Accepted August 23, 2022; Published September 15, 2022

Abstract. An implicit assumption in the original version of Holditch’s theorem is $C^1$ -regularity and strict convexity of the envelope generated by a chord traveling around a convex curve $\mathbb C$ . We establish that this holds when $\mathbb C$ is $C^2$ -regular with positive curvature and the chordlength is sufficiently small. We also consider the case where $\mathbb C$ is polyhedral. Then, strict convexity of the envelope may not hold, but, for sufficiently small chordlength, it is nevertheless $C^1$ -regular. The case of a general convex curve $\mathbb C$ remains an open problem.

How to Cite this Article:
H. Proppe, A. Stancu, R.J. Stern, Holditch’s Envelope, Commun. Optim. Theory 2022 (2022) 13.