On the Center of Surface Area of the Boundary of a Star-Shaped Region

Abstract

AbstractEvery line passing through the center of a circular disk bisects its area. Not every planar region has a point with this property. It turns out that if a compact connected region is star-shaped at its center of area, then it must be centrally symmetric. However, there exists a non-centrally symmetric star-shaped region whose boundary has a center of length, which is a point where every line passing through this point cut its length in half. In this article, we characterize such regions and provide a method to create them. Additionally, we provide new examples and a method to generate analogous objects in higher dimensions. AcknowledgmentThe author is extremely grateful to Professor Thomas F. Banchoff for introducing the research problem and for his invaluable support and guidance during the initial investigation. The author thanks the referees for their insightful comments and valuable suggestions. The author is partially supported by Faculty of Science, Srinakharinwirot University through Grant 170/2563.0090`Additional informationNotes on contributorsThunwa Theerakarn Thunwa Theerakarn (thunwa@g.swu.ac.th) has been teaching at Srinakharinwirot University in Bangkok, Thailand since 2019. He received Ph.D. in mathematics from the University of California, Berkeley. He received M.Sc. in applied mathematics and B.Sc. in mathematics from Brown University.