Convexity of a set on the plane in a given direction (Q1776310)

The authors consider twelve different conditions which might be used to define convexity of a subset \(Q\) of \(\mathbb{C}\) in a given direction \(\beta \in \mathbb{C}\backslash \{0\}\). The first of these conditions reads ``for any \(\alpha \in \mathbb{C}\), the set \((\alpha -\beta \mathbb{C} _{+})\backslash Q\) is arcwise connected'', where \(\mathbb{C}_{+}\) denotes the half-plane Re \(z>0\). The other conditions arise by replacing ``arcwise connected'' by ``connected'' or ``a semicontinuum'', by replacing \(\mathbb{C}_{+}\) by its closure \(\overline{\mathbb{C}}_{+}\), or by replacing set-theoretic difference by intersection. The authors prove various implications among these conditions for general subsets as well as for bounded or connected subsets. They also prove that the obtained set of implications is complete. Furthermore, four variants of convexity in a given direction with respect to a given point are considered, and they are proved to be equivalent. There is also a section on ``intrinsic'' definition of convexity in direction, which contains for instance the following theorem: ``For convexity of a set \(Q \) in direction \(\beta \), it is necessary that each of its connected components is convex in direction \(\beta \).'' (Here the semicontinuum version of the above-mentioned condition is used.) The final sections contain a series of properties of convexity in direction, concerning for instance the set of all directions \(\beta \) such that \(Q\) is convex in direction \(\beta \) with respect to a given point \(\alpha \), or the convex hull of \(Q\) with respect to a set of directions.