On decompositions of compact convex sets (Q1392727)

The author considers a concept of convexity given by: ``A system \(\mathcal C \subset \) \(2^X\) is called a weak convexity on the set \(X\) if the intersection \(\cap C'\) of any nonempty subsystem \(C' \subset \mathcal C\) belongs to \(\mathcal C\)''. A bijection \(\tau : X \rightarrow X\) is said to be \(\mathcal C\)-invariant if \(\tau(\mathcal C)\) and \(\tau^{-1}(\mathcal C) \subset \mathcal C\). Firstly, the author proves by induction that in a topological space X, if a compact convex set \(C\in \mathcal C\) admits a decomposition \(C=A \cup B , \tau(A)=B\), where \(\tau\) is a \(\mathcal C\)-invariant homeomorphism, then there exists a nonempty compact convex subset \(\widetilde{C} \subset C\) such that \(\tau(\widetilde{C})=\widetilde{C}\). Using this fixed point property, there are proved three theorems of indecomposability of a compact convex set: -- in a locally convex topological linear space under the group of affine homeomorphisms [see \textit{M. Edelstein}, Proc. Am. Math. Soc. 115, No. 3, 737-739 (1992; Zbl 0776.46009)]; -- in the real line \(\mathbb E^1\) under the group of homeomorphisms of \(\mathbb E^1\) onto itself; -- in the \(d\)-dimensional sphere \(S^d\) or \(d\)-dimensional hyperbolic space \(H^d\) under the group of isometries of \(S^d\) or \(H^d\). Some open problems are formulated: the \(n\)-divisibility for \(n \geq 3\), if the condition of convexity can be weakened or, if the theorems can be formulated for ``sets being homeomorphic to compact convex sets''.