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An analogue of the Krein-Milman theorem for star-shaped sets. (English) Zbl 1043.52006

In view of the well-known Krein-Milman theorem, every compact convex subset \(K\) of \(\mathbb R^n\) is the convex hull of the set \(\text{ext\,} K\) of its extreme points. The authors prove an analogue of that theorem for the class of compact star-shaped sets (Theorem 2.7 below). Let \(S\) be a compact star-shaped set in \(\mathbb R^n\) and \(K\) a nonempty compact convex subset of its kernel. A point \(q_0 \in S \setminus K\) is an extreme point of \(S\) modulo \(K\) if and only if \[ \forall p \in S \setminus (K \cup \{q_0\}) \;q_0 \notin \text{conv}(K \cup \{p\}). \] For any \(K \subset \mathbb R^n\) and \(E:= \mathbb R^n \setminus K\), a closure operator \(\sigma_K: {\mathcal P}(E) \to {\mathcal P}(E)\) is defined. Further, \(\tau_K(A): = K \cup \sigma_K(A)\) for every \(A \subset E.\) Main result is the following. Theorem 2.7. The set \(S_0\) of extreme points of \(S\) modulo \(K\) satisfies \(\tau_K(S_0)=S.\) If, moreover, \(S' \subset S \setminus K\) satisfies \(\tau_K(S')=S,\) then \(S_0 \subset S'\).

MSC:

52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A37 Other problems of combinatorial convexity
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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