Properties of lexicographic maximum of convex compact (Q2759395)

This paper deals with properties of lexicographic maximum of convex compacts in \(\mathbb{R}^{n}\). Let \(X\) be a convex compact in \(\mathbb{R}^{n}\) and let \(X=X^1\cap X^2\cap X^3\), where \(X^{i}, i=1,2,3\) is a convex compact in \(\mathbb{R}^{n}\). Let us denote \(x^{i}=\mathop{\max}\limits^{L} X^{i}, i=1,2,3\), and \(Y^1=X^1\cap X^2, Y^2=X^1\cap X^3, Y^3=X^2\cap X^3\), \(y^{i}=\mathop{\max}\limits^{L} Y^{i}, i=1,2,3\). If \(y^1\mathop{\leq}\limits^{L} y^2\mathop{\leq}\limits^{L} y^3\) and \(y^{*}=\mathop{\max}\limits^{L} X\), then in \(\mathbb{R}^2\) we have \(y^{*}=y^1\). A method of searching the lexicographic maximum for the convex compacts is proposed.