Minimal pairs of convex compact sets
The author considers the equivalence relation \((A,B) \sim (C,D) \Leftrightarrow A + D = B + C\), and the ordering \((A,B) \leq (C,D) \Leftrightarrow A \subseteq C\) and \(B \subseteq D\) in the collection of pairs of convex compact sets in \(\mathbb{R}^ 2\) and \(\mathbb{R}^ 3\). He proves that in the case of \(\mathbb{R}^ 2\) a minimal pair in any given equivalence class is unique up-to-translation. In order to prove it he introduces nondecreasing ``boundary length functions'' \(f_ A : [0,2 \pi] \to \mathbb{R}_ +\) for every convex compact set \(A\). These functions have the following properties: \(f_{A + B} = f_ A + f_ B\), \(f_{aA} = af_ A\), and \(f_ A = f_ B\) if and only if \(B\) is a translation of \(A\). The ``minimality'' of a pair of convex compact sets is characterized with the help of these functions. The author gives an example of two equivalent minimal pairs of convex compact sets in \(\mathbb{R}^ 3\) which are not translations of each other.
scientific article; zbMATH DE number 687060