Visibility in semi-convex spaces
A semi-convex space is a pair \((X,C_X)\) consisting of a set \(X\) and a collection \(C_X\) of subsets of \(X\) such that \(\varnothing\in C_X\) and, for all nonempty \(C \subseteq C_X\), one has \(\cap C\in C_X\) (in contrast with the definition of a convexity space, it is not required that \(X\in C_X)\). The convex hull of \(Y\subseteq X\) is defined as \(C_X- \text{hull} (Y)=\cap \{C\in C_X| Y\subseteq C\}\), with the unusual convention \(\cap \varnothing= \varnothing\). Setting \(C_Y= \{C\in C_X| C\subseteq Y\}\), one says that \(x\) \(C_Y\)-sees \(y\) in \(Y\) if \(C_Y-\text{hull} (\{x,y\}) \neq\varnothing\). One further defines \(C_X\)-kernel\((Y)= \{y\in Y| \text{ for all } y'\in Y,\;y\) \(C_Y\)-sees \(y'\}\). The maximal elements (with respect to inclusion) of \(C_Y\) are called \(C_X\)-skulls of \(Y\). The authors prove several theorems relating skulls and kernels, and study semi-convex spaces in the plane induced by metrics. The theory is illustrated with two examples of semi-convex spaces based on geodesic convexity and staircase convexity.
scientific article; zbMATH DE number 1106610
scientific article