Minimal pairs representing selections of four linear functions in \(\mathbb{R}^3\) (Q2718682)

Let \(f,f_{1},\ldots ,f_{m}:U\rightarrow R\) be continuous convex functions, where \(U\subset R^{n}\) is an open set; \(f\) is a continuous selection of \( f_{1},\ldots ,f_{m}\) if \(I(x):=\{i\in \{1,\ldots ,m\}\mid f_{i}(x)=f(x)\}\) is nonempty for every \(x\in U,\) the class of such \(f\) being denoted by \( CS(f_{1},\ldots ,f_{m}).\) NEWLINENEWLINENEWLINE\textit{S. G. Bartels, L. Kuntz} and \textit{S. Scholtes} [Nonlinear Anal., Theory Methods Appl. 24, No. 3, 385-407 (1995; Zbl 0846.49005)] showed that every continuous selection of the linear functions \(l_{1},\ldots ,l_{m}\) on \(R^{n}\) has a representation of the form \(l(x)=\min_{i\in \{1,\ldots ,r\}}\max_{j\in M_{i}}l_{j}(x),\) and this representation is unique provided \(l_{1},\ldots ,l_{m}\) are affinely independent. The authors observe that every such function \(l\) can be represented as \(p_{A}-p_{B}\) with \(A,B\) compact convex subsets of \(R^{n},\) \(p_{A}\) being the support function of \(A.\) In the paper one states that \(CS(y_{1},y_{2},y_{3},-y_{1}-y_{2}-y_{3})\) for \( n=3\) consists of \(166\) continuous selections which are represented by \(16\) essentially different minimal pairs \([A,B].\)