Staircase kernels for orthogonal \(d\)-polytopes
The author considers special \(d\)-polytopes in connection with a special kind of starshapeness. The result is a generalization of a theorem for 2-polytopes given in a former note by the author [J. Geom. 51, No. 1-2, 22-30 (1994; Zbl 0815.52004)]. \textit{L. Danzer}, \textit{B. Grünbaum} and \textit{V. Klee}, Proc. Symp. Pure Math. 7, 101-180 (1963; Zbl 0132.17401)] had discussed ordinary starshaped sets with the help of visibility via straight line segments. Now, following \textit{R. Motwani}, \textit{A. Raghunathan} and \textit{H. Saran} [J. Comput. Syst. Sci. 40, No. 1, 19-48 (1990; Zbl 0705.68082)], the author uses visibility via staircase paths. An (oriented) path with edges parallel to the coordinate axes will be called a staircase path if no two of its edges have opposite directions. Then for two points \(x\) and \(y\) in a polytope \(S\) one says \(x\) is visible from \(y\) via staircase paths if there is a staircase path in \(S\) which contains both \(x\) and \(y\). Sharper, \(x\) is clearly visible from \(y\) via staircase paths if for some neighborhood \(N\) of \(x\) each point of \(S\) which belongs to \(N\) is visible from \(y\) via staircase paths. The polytope \(S\) in question is an orthogonal \(d\)-polytope, that is a connected union of finitely many convex polytopes, whose edges are parallel to the coordinate axes. Such a polytope is starshaped via staircase paths if for some point \(p\) in \(S\) each point of \(S\) is visible from \(p\) via staircase paths. The set of all such points \(p\) is called the staircase kernel of \(S\). For these staircase kernels of orthogonal \(d\)-polytopes the author gives a simple representation using clearly visibility. Furthermore she shows how to change the result by using only visibility. Namely, for clearly visibility it suffices to consider only those points of \(S\) being points of local nonconvexity, while only visibility needs all the points of the boundary of \(S\).
scientific article; zbMATH DE number 920726