## 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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