Some generalizations of the shadow problem in the Lobachevsky space. (English. Ukrainian original) Zbl 1483.52008

Kostin, A. V.

Ukr. Math. J. 73, No. 1, 67-75 (2021); translation from Ukr. Mat. Zh. 73, No. 1, 61-68 (2021).

A set \(U\) in the \(n\)-dimensional Euclidean space \(E^n\) is called \(m\)-convex with respect to a point \(P \notin E^n \setminus U\) for some \(1 \leq m < n\) if there is no \(m\)-dimensional subspace containing \(P\) and disjoint from \(U\), and \(U\) is called \(m\)-semiconvex with respect to \(P\) if there is no \(m\)-dimensional half subspace containing \(P\) and disjoint from \(U\). The shadow problem asks for the minimum number \(N\) of open/closed balls mutually disjoint and disjoint from a point \(P\) such that their union is \(1\)-convex with respect to \(P\) and their centers lie on the same sphere centered at \(P\).
The aim of the author is to present solutions for variants of this problem in the hyperbolic space \(H^n\). After defining \(m\)-convexity and \(m\)-semiconvexity analogously to the Euclidean definition, he examines variants of the shadow problem in the \(2\)- and \(3\)-dimensional hyperbolic space. More specifically, he proves that the minimum number of open (closed) disjoint disks in the hyperbolic plane whose centers are at the same distance from some point \(P\), are disjoint from \(P\) and their union is \(1\)-convex with respect to \(P\), is \(2\), and the same holds if we replace disks by horodisks. Furthermore, for both open (closed) disks or horodisks this number is \(3\) if we replace \(1\)-convexity by \(1\)-semiconvexity.
For points in the \(3\)-dimensional hyperbolic space the author remarks that analogously to the Euclidean problem it can be shown that the minimum number of pairwise disjoint open (closed) balls in \(H^3\) whose centers are on a sphere centered at a point \(P\), and are disjoint from \(P\) and their union is \(1\)-convex with respect to \(P\), is \(4\). Furthermore, he proves that if we drop the condition that the centers of the balls lie on a sphere centered at \(P\), or assume that they lie on an ellipsoid centered at \(P\), this number is \(3\). He also proves results about the case when some or all of the balls are replaced by horoballs.

Reviewer: Zsolt Lángi (Budapest)

MSC:

52A55	Spherical and hyperbolic convexity
52A30	Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
51M10	Hyperbolic and elliptic geometries (general) and generalizations

Keywords:

hyperbolic space; shadow problem; horoball; \(m\)-convexity; \(m\)-semiconvexity

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