On the isoperimetric inequalities for Reuleaux polygons
The authors prove two classical results about Reuleaux polygons in the Euclidean plane, namely the Blaschke-Lebesgue Theorem and the Firey-Sallee Theorem, both of which give isoperimetric inequalities for Reuleaux polygons. A Reuleaux polygon is a compact convex subset \(C\) of \({\mathbb R}^2\) of constant width which admits a finite pinching set. A pinching set \(V\) of \(C\) is a subset of the boundary \(\text{bd}(C)\) of \(C\) such that each diameter of \(C\) is incident with at least one point of \(V\). The order of \(C\) is defined as follows. For \(v \in \text{bd}(C)\) let \(N_C(v) := \{w \in \text{bd}(C)\mid\|v-w\|= \text{diam}(V)\}\) and put \(V_\infty(C) := \{v \in \text{bd}(C)\mid N_C(v)\) is a non-degenerate arc\} as the set of vertices of \(C\). The cardinality of the set of vertices of \(C\) is called the order of \(C\). The Blaschke-Lebesgue Theorem states that the Reuleaux triangle of width 1 has minimal area among all Reuleaux polygons of width 1. The Firey-Sallee Theorem characterizes the regular Releaux polygons of order \(n\) as those Releaux polygons of order at most \(n\) that have maximal area. The authors prove both theorems in a unified manner using continuity arguments in compact spaces. Moreover, the authors describe a construction that turns a given Reuleaux polygon into another Reuleaux polygon (called the \(\Lambda\)-variant), which is generically of equal or lower order than the original one. However, for certain cases it can be shown that the order of the \(\Lambda\)-variant has increased by two. Based on this observation they prove another result of Blaschke, namely that for any Reuleaux polygon \(C\) of order \(n\geq 5\) there is a \(\Lambda\)-variant of \(C\) of order \(n-2\) that has a smaller area than \(C\).
scientific article; zbMATH DE number 1508004