塵封的經典：初等數學經典文獻選讀(第一卷) （簡體書）

《歐美初等數學經典系列（第1輯）．塵封的經典：初等數學經典文獻選讀（第1卷）》搜集初等數學的經典文獻，包括“拉格雷舊成果的新運用”“平面對成群的識別與標記”“匈牙利的數學發展”“Bonnesen等周不等式”“准割圓多項式”“n次冪差分的歐拉公式”“算數級數”“三角不等式”“調和級數的一些收斂子級數”等在內，編輯成書，便於讀者進行學習和查閱，《歐美初等數學經典系列（第1輯）．塵封的經典：初等數學經典文獻選讀（第1卷）》適用于學生學習同時也可作為數學愛好者的興趣讀物。．

《塵封的經典:初等數學經典文獻選讀(第1卷)》適用于學生學習同時也可作為數學愛好者的興趣讀物。

All of these results were for convex curves only,and the extension to non-convex curves required essentially new methods.
The first results are due to Erhard Schmidt in 1939.Using analytic rather than geometric methods,he derives several Bonnesen-type inequalities for plane domains bounded by an arbitrary rectifiable Jordan curve([68,p.690-694]).He does not,however,obtain the inequalities of Theorems 1 and 2 above.The first method to succeed here was integral geometry.The book of Blaschke(p. 26)gives a proof of (11)and(16)for convex curves,due to Santalo.Also using integral geometry,Hadwiger in 1941[41]obtained results equivalent to inequalities(15)and(20) for arbitrary rectifiable Jordan curves.He does not appear to notice the connections with Bonnesen inequalities, however,until a later paper[42],where he derives the inequalities(12),(13),(17),(18),(22)and(23),but only for convex domains.
In the meanwhile,in the same volume of the journal that contains the first of Hadwiger's papers,there appeared a fundamental paper of Fiala.In it Fiala develops another method for proving Bonnesen inequalities for non-convex curves.That is the method of interior parallels,and,except for the proof of Theorem 3 above,it is the method used here.Fiala's principal focus is on obtaining isoperimetric inequalities on curved surfaces (see Section B below),but his paper applies in particular to the plane and is the first to give explicitly(on p.336)(11)and(14) for non-convex curves.His proof is for analytic Jordan curves.One could then obtain the result for more general curves by approximation.