On iterative processes generating dense point sets

Summary

The central problem of this paper is the question of denseness of those planar point sets <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathcal{P}$, not a subset of a line, which have the property that for every three noncollinear points in $\mathcal{P}$, a specific triangle center (incenter (IC), circumcenter (CC), orthocenter (OC) resp.) is also in the set $\mathcal{P}$. The IC and CC versions were settled before. First we generalize and solve the CC problem in higher dimensions. Then we solve the OC problem in the plane essentially proving that $\mathcal{P}$ is either a dense point set of the plane or it is a subset of a rectangular hyperbola. In the latter case it is either a dense subset or it is a special discrete subset of a rectangular hyperbola.