Subsets of the square with the continuous and order preserving fixed point property (Q412733)

Let \(K\) be a convex subset contained in the interior of the unit square \(I:=[0,1]\times [0,1]\) and \(P=I\setminus K\). Further, denote \(g=\inf(K)\), \(l=\sup(K)\); and consider the conditions: (A1): SE-border is a nonempty closed set containing \(SE-boundary\cap \text{int}[g,l]\), (A2): NW-border is a nonempty closed set containing \(NW-boundary\cap \text{int}[g,l]\), (A): the intersection between (A1) and (A2), (B): at least one of \(g\) or \(l\) is in \(P\), (non-B): the negation of (B), (C1): one of the sets \(SE_n\cap NW-border\) or \(NW_n\cap SE-border\) is nonempty, (C2): either \(\Pi_1(g_n)> \Pi_1(l^n)\), \(\Pi_2(l_n)\geq \Pi_2(g^n)\), or \(\Pi_1(g_n)\leq \Pi_1(l^n)\), \(\Pi_2(l_n)< \Pi_2(g^n)\).NEWLINENEWLINEThe following is the main result of the paper:NEWLINENEWLINE\textbf{Theorem.} The subset \(P\) has the cop fixed point property if and only if either (A) and (B) or (A) and (non-B) holds; where, in the second case, either (C1) or (C2) must be true.NEWLINENEWLINEHere, \(\Pi_i\) is the projection map onto the \(i\)-th coordinate (where \(i\in \{1,2\}\)).