Discrete and lexicographic Helly-type theorems
Helly's theorem says that if every \(d+1\) elements of a given finite family of convex sets in \(\mathbb{R}^{d}\) have a common point, then there is a point common to all sets of the family. The author defines three new types of Helly theorems: ``discrete'' Helly theorems -- where the common point should belong to an a-priori given set, lexicographic Helly theorems -- where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. A typical example of a discrete Helly theorem is the following theorem due to \textit{J.-P. Doignon} [J. Geometry 3, 71--85 (1973; Zbl 0245.52004)]: ``Let \(H\) be a finite family of at least \(2^{d}\) convex sets in \(\mathbb{R}^{d}\). If every \(2^{d}\) or fewer members of \(H\) have a common point with integer coordinates, then there is a point with integer coordinates common to all members of \(H\).'' The author discusses and proves a series of theorems of the above mentioned types, including piercing and transversal problems.
scientific article; zbMATH DE number 5313107
scientific article