Cages of Small Length Holding Convex Bodies
- PDF / 708,406 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 16 Downloads / 230 Views
Cages of Small Length Holding Convex Bodies Augustin Fruchard1 · Tudor Zamfirescu2,3,4 Received: 8 August 2017 / Revised: 22 July 2019 / Accepted: 4 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract A cage G, defined as the 1-skeleton of a convex polytope in 3-space, holds a compact set K if G cannot move away without meeting the relative interior of K . The main results of this paper establish the infimum of the lengths of cages holding various compact convex sets. First, planar graphs and Steiner trees are investigated. Then the notion of points almost fixing a convex body in the plane is introduced and studied. The last two sections treat cages holding 2-dimensional compact convex sets, respectively the regular tetrahedron. Keywords Immobilisation · Skeleton · Steiner tree · Convex body Mathematics Subject Classification 52A15 · 52A40 · 52B10
1 Introduction A cage is the 1-dimensional skeleton of a 3-dimensional convex polytope in R3 . A compact set G is said to hold another compact set K if G is disjoint from the relative interior of K and if G cannot be rigidly moved to a position far away without intersecting the relative interior of K on its way. Here, a move means a continuous path starting from the identity in the space of isometries of R3 . “Far away” means that Dedicated to the memory of Ricky Pollack. Editor in Charge: János Pach Augustin Fruchard [email protected] Tudor Zamfirescu [email protected] 1
Département de Mathématiques, IRIMAS, Université de Haute-Alsace, 68093 Mulhouse, France
2
Fakultät für Mathematik, Technische Universität Dortmund, 44221 Dortmund, Germany
3
Institute of Mathematics “Simion Stoilow”, Romanian Academy, 010702 Bucharest, Romania
4
College of Mathematics, Hebei Normal University, 050024 Shijiazhuang, P. R. China
123
Discrete & Computational Geometry
the convex hulls of K and of the moved G are disjoint. Already in 1959, Coxeter [5] raised the problem of finding the infimum of the total lengths of cages holding the ball of radius 1 in R3 . In the following years, Besicovitch [2] and Aberth [1] solved Coxeter’s problem. In the present paper, we extend the investigation to other compact convex sets replacing the ball. √ The space R3 is endowed with its Euclidean norm x = x| x, where · | · is the usual scalar product. For distinct x, y ∈ R3 , let x y be the line through x, y and x y the line-segment from x to y. The open line-segment x y \ {x, y} is denoted by ]x y[. Given a line x y oriented from x to y, (x y)+ denotes the open half-plane on the left of x y. The measure of an angle x yz is denoted by ∠x yz. The measure of the angle between two lines or planes X and Y is denoted by ∠(X , Y ). These angles will be oriented if the context is in the plane, and unoriented if the context is in 3-space. As usual, for M ⊂ Rd with d ≥ 2, the convex hull conv M of M is the intersection of all convex subsets of Rd containing M, and its affine hull aff M is the intersection of all affine subspaces of Rd containing M.
Data Loading...
