Journal of Convex Analysis 32 (2025), No. 3, 723--730
Copyright Heldermann Verlag 2025
Approximation Approach to Holditch's Theorem
Harald Proppe
Dept. of Mathematics and Statistics, Concordia University, Montreal, Canada
hal.proppe@concordia.ca
Alina Stancu
Dept. of Mathematics and Statistics, Concordia University, Montreal, Canada
alina.stancu@concordia.ca
Ronald J. Stern
Dept. of Mathematics and Statistics, Concordia University, Montreal, Canada
ron.stern@concordia.ca
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.
Keywords: Holditch's theorem, convex analysis, Broman's condition, local obtuseness, smooth approximation.
MSC: 26A27, 49J52.
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