On the smallest disk containing a planar reduced convex body
A convex body \(R\subseteq E^d\) is reduced if for every convex body \(P\subseteq R\), \(P\) different from \(R\), \(\Delta (P)< \Delta (R)\), where \(\Delta (C)\) is the thickness of the convex body \(C\subseteq E^d\), i.e. the minimum width of \(C\). The author proves that every reduced convex body \(R\subset E^2\) is contained in a disk of radius \({1\over 2}\sqrt{2}\Delta (R)\). This estimate is sharp when \(R\) is a quarter of a disk, i.e. any set \(Q\) obtained when we dissect an arbitrary disk by two perpendicular diagonals into four congruent closed sets. Moreover, the author shows that every reduced polygon \(R\) is contained in a disk of radius \({2\over 3}\Delta (R)\). This estimate is sharp only for regular triangles. It is also proved in the paper that for \(d\geq 3\) there are reduced bodies with arbitrary large finite ratio of the diameter to the thickness. This shows that there is no analogous property in \(d\geq 3\) dimensions.
scientific article; zbMATH DE number 1993488