The \(\varepsilon\)-positive center set and its applications
The well-known Bonnesen inequality states that the so called Bonnesen functional \(B(r)=L\,r-A-\pi r^2\) of a plane convex body \(K\) of area \(A\) and perimeter \(L\) is non-negative when \(r\) lies between the inradius and circumradius of \(K\). For a given point \(c\in K\) let \(r_c\) (\(R_c\)) be the maximal (minimal) radius of a disk centered at \(c\) contained (containing) \(K\). A point \(c\in K\) is called an \(\epsilon\)-center if \(B(r_c), B(R_c)\geq \epsilon\). For \(\epsilon=0\) it is called a positive center and was introduced by \textit{M. E. Gage} [Proc. Am. Math. Soc. 110, No. 4, 1041--1048 (1990; Zbl 0725.52003)]. He showed that the center of the minimal annulus is a positive center and in this way he strengthened an isoperimetric type inequality of Bonnesen. This more general concept of \(\epsilon\)-centers goes back to \textit{M. J. Kaiser} [Appl. Math. Lett. 9, No. 6, 67--70 (1996; Zbl 0877.52004)] and he conjectured, among others, that the set of all \(\epsilon\)-centers is convex. In the paper under review the authors verify the conjecture.
scientific article; zbMATH DE number 6688737