A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck

• Published in: Inventiones mathematicae Vol. 129; no. 1; pp. 141 - 212
• Main Author: Rychlik, Marek R.
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01-07-1997
• ISSN: 0020-9910; 1432-1297
• DOI: 10.1007/s002220050161

The Equichordal Point Problem can be formulated in simple geometric terms. If (ProQuest: Formulae and/or non-USASCII text omitted; see image) is a Jordan curve on the plane and (ProQuest: Formulae and/or non-USASCII text omitted; see image) then the segment (ProQuest: Formulae and/or non-USASCII text omitted; see image) is called a chord of the curve (ProQuest: Formulae and/or non-USASCII text omitted; see image) . A point inside the curve is called equichordal if every two chords through this point have the same length. The question was whether there exists a curve with two distinct equichordal points (ProQuest: Formulae and/or non-USASCII text omitted; see image) and (ProQuest: Formulae and/or non-USASCII text omitted; see image) . The problem was posed by Fujiwara in 1916 and independently by Blaschke, Rothe and Weizenböck in 1917, and since then it has been attacked by many mathematicians. In the current paper we prove that if (ProQuest: Formulae and/or non-USASCII text omitted; see image) and (ProQuest: Formulae and/or non-USASCII text omitted; see image) are two distinct points on the plane and (ProQuest: Formulae and/or non-USASCII text omitted; see image) is a Jordan curve such that the bounded region (ProQuest: Formulae and/or non-USASCII text omitted; see image) cut out by (ProQuest: Formulae and/or non-USASCII text omitted; see image) is star-shaped with respect to both (ProQuest: Formulae and/or non-USASCII text omitted; see image) and (ProQuest: Formulae and/or non-USASCII text omitted; see image) then (ProQuest: Formulae and/or non-USASCII text omitted; see image) is not equichordal. The original question was posed for convex (ProQuest: Formulae and/or non-USASCII text omitted; see image) , and thus we have solved the Equichordal Point Problem completely. Our method is based on the observation that (ProQuest: Formulae and/or non-USASCII text omitted; see image) would be an invariant curve for an algebraic map of the plane. It would also form a heteroclinic connection. We complexify the map and obtain a multivalued algebraic map of (ProQuest: Formulae and/or non-USASCII text omitted; see image) . We develop criteria for the existence of heteroclinic connections for such maps.[PUBLICATION ABSTRACT]