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Inscribed square problem
The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: ''Does every plane simple closed curve contain all four vertices of some square?'' This is known to be true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911.〔Toeplitz, O. : "Ueber einige Aufgaben der Analysis situs" ''Verhandlungen der Schweizerischen Naturforschenden Gesellschaft in Solothurn'', 94 (1911), p. 197.〕 Some early positive results were obtained by Arnold Emch〔.〕 and Lev Schnirelmann.〔.〕 As of 2015, the general case remains open.〔.〕
== Problem statement ==
Let ''C'' be a Jordan curve. A polygon ''P'' is inscribed in ''C'' if all vertices of ''P'' belong to ''C''. The inscribed square problem asks:
: ''Does every Jordan curve admit an inscribed square?''
It is ''not'' required that the vertices of the square appear along the curve in any particular order.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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