On iterative processes generating dense point sets (Q2456230)

In this paper the authors obtain two theorems concerning denseness of some point sets. Theorem 1: Let \(\mathcal{P}\) be a point set in \(E^n\) containing a subset of \(n+1\) affinely independent points. If there exists a number \( 0\leq \varrho < 1\) such that for any ball \(B\) determined by \(n+1\) affinely independent points of \(\mathcal{P}\) the ball \(\varrho B\) contains at least one point of \(\mathcal{P}\), then \(\mathcal{P}\) is dense in \(E^n\). Theorem 2: Let \(\mathcal{P}\) be a planar set containing at least three noncollinear points with the property that the orthocenter of any triangle determined by three noncollinear points in \(\mathcal{P}\) is also in \(\mathcal{P}\). Then exactly one of the following holds true: (i) \(\mathcal{P}\) consists of the vertices of a triangle together with its orthocenter. (ii) \(\mathcal{P}\) is a discrete infinite subset of a rectangular hyperbola, parametrizable by two parameters. (iii) \(\mathcal{P}\) is a dense subset of a rectangular hyperbola. (iv) \(\mathcal{P}\) is a dense point set in the plane.