On a Convex Operator for Finite Sets

Let $S$ be a finite set with $n$ elements in a real linear space. Let \cJ_S be a set of $n$ intervals in \nR. We introduce a convex operator \co(S,\cJ_S) which generalizes the familiar concepts of the convex hull \conv S and the affine hull \aff S of $S$. We establish basic properties of this operator. It is proved that each homothet of \conv S that is contained in \aff S can be obtained using this operator. A variety of convex subsets of \aff S can also be obtained. For example, this operator assigns a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For \cJ_S which consists of bounded intervals, we give the upper bound for the number of vertices of the polytope \co(S,\cJ_S).Comment: 20 pages, 16 figure

Holes or Empty Pseudo-Triangles in Planar Point Sets

Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ and $E(5, \ell)$, and prove bounds on $E(k, 6)$ and $E(6, \ell)$, for $k, \ell\geq 3$. By dropping the emptiness condition, we define another related quantity $F(k, \ell)$, which is the smallest integer such that any set of at least $F(k, \ell)$ points in the plane, no three on a line, contains a convex polygon with $k$ vertices or a pseudo-triangle with $\ell$ vertices. Extending a result of Bisztriczky and T\'oth [Discrete Geometry, Marcel Dekker, 49--58, 2003], we obtain the exact values of $F(k, 5)$ and $F(k, 6)$, and obtain non-trivial bounds on $F(k, 7)$.Comment: A minor error in the proof of Theorem 2 fixed. Typos corrected. 19 pages, 11 figure