Some inequalities for planar convex figures
The paper concerns inequalities for compact convex subsets of the Euclidean plane. Let \({\mathcal K}^2\) be the class of such sets. For every \(K \in {\mathcal K}^2\) let \(A(K)\), \(D(K)\), and \(r(K)\) be the area, the diameter, and the inradius of \(K\). The authors prove the following. Theorem 1.1. For every \(K \in {\mathcal K}^2\) \(A(K) \leq 2D (K) \cdot r(K)\). The equality holds if and only if \(\text{int} K = \emptyset\). The term ``width'' is used here in a sense different than usually. It is understood as thickness, i.e. the infimum of widths in all directions.
scientific article; zbMATH DE number 698834