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一猜想的提出夏道行在中学数学课外读物《π和 e》一书中,为了证明圆的周长和它的直径的比是一常数——π,用极限语言给周长下了一个定义:圆内接多边形当边效无限增加时其周长的极限即是圆周长.但所得序列的极限是否存在呢?需要证明序列的有界性.读物仅证明了一特殊情况——圆外切正方形的周长大于其内圆内接正八边形的周长(见图1),对于更一般的情况,自然产生一个猜想:凸多边形的周长大于其内任一凸多边形的周长.如此命题真,显然序列有界性成立,将其极限定义为圆周
A Conjecture Proposed Xia Daoxing In the book “π and e” outside the middle school math class, in order to prove that the ratio of the circumference of a circle to its diameter is a constant, π, the limit language is used to define the perimeter: Circle Inscribed Polygons When the edge increases infinitely, the limit of its perimeter is the circumference. However, does the limit of the obtained sequence exist? It is necessary to prove the boundedness of the sequence. The reading only proves a special case - the circle cut square The circumference of the circle is greater than the perimeter of the inner circle that is inscribed with a regular octagon (see Figure 1). For the more general case, a conjecture naturally arises: the perimeter of a convex polygon is greater than the perimeter of any convex polygon within it. True, obviously the sequence boundedness is established and its limit is defined as the circumference