On extreme convex subsets of the plane (Q1271921)

If \(Q\) is a convex compact subset in \({\mathbb R}^2\) with non-empty interior, let \(\Lambda\) be the family of all non-empty closed convex subsets of \(Q\). As for \(A\), \(B \in \Lambda\), \(\alpha \in [0,1]\) \[ \alpha A+(1-\alpha) B=\{\alpha a +(1-\alpha) b : a \in A,\;b \in B \} \in \Lambda, \] \(\Lambda\) is convex. Therefore it is a natural question to ask what are the extreme elements of \(\Lambda\). Characterizations of those had been known in the special case when \(Q\) is strictly convex [\textit{R. Grząślewicz}, Arch. Math. 43, 377-380 (1984; Zbl 0536.52002)]. The conditions given there are not sufficient for a characterization of the extreme elements in the general case. In this paper the author gives a nice geometric characterization of the extreme elements in the general case.