×

\(C\)-minimal pairs of compact convex sets. (English) Zbl 0886.52002

Let \(K(X)\) denote the set of all nonempty compact convex subsets of a real topological vector space \(X\). The pair \((A,B)\in K^2(X) =K(X) \times K(X)\) is called convex if \(A \cup B\) is convex. It is called \(C\)-minimal if \((A+C, B+C)\) is convex and \((A+C_1, B+C_1)\) is not convex for any proper subset \(C_1\in K(X)\) of \(C\). In the one and two-dimensional cases, \((A,B)\) is both \(C_1\)-minimal and \(C_2\)-minimal only if \(C_2= C_1+x\) for some vector \(x\). Two pairs \((A,B)\), \((C,D)\in K^2(X)\) are said to be equivalent if \(A+D= B+C\). One says that \((C,D)\) is a convex hull of \((A,B)\) if it is convex, \((A,B)\) and \((C,D)\) are equivalent, one has \(A\subseteq C\) and \(B\subseteq D\), and for any convex pair \((C_1, D_1)\) equivalent to \((A,B) \) with \(A\subseteq C_1\subseteq C\) and \(B\subseteq D_1\subseteq D\) it follows that \((C,D)= (C_1,D_1)\).
The existence of a convex hull for every pair is proved. Moreover, several criteria for convexity preserving reduction of a convex pair of convex sets within its equivalence class are presented. Application to minimal representations of sub- and superdifferentials of quasidifferentiable functions are also discussed.

MSC:

52A07 Convex sets in topological vector spaces (aspects of convex geometry)
26B25 Convexity of real functions of several variables, generalizations
90C30 Nonlinear programming
PDFBibTeX XMLCite
Full Text: EuDML EMIS