4 research outputs found
ARRANGEMENTS OF HOMOTHETS OF A CONVEX BODY II
A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most 2 . 3(d) members. This improves a result of Polyanskii (Discrete Mathematics 340 (2017), 1950-1956). Using similar ideas, we also give a proof the following result of Polyan- skii: Let , K-1, ... ,K-n be a sequence of homothets of the o-symmetric convex body K, such that for any i < j, the center of K-j lies on the boundary of K-i. Then n = O(3(d)d)
Arrangements of homothets of a convex body II
A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a ddimensional convex body has at most 2 · 3d members. This improves a result of Polyanskii (Discrete Mathematics 340 (2017), 1950–1956). Using similar ideas, we also give a proof the following result of Polyanskii: Let K1, . . . , Kn be a sequence of homothets of the o-symmetric convex body K, such that for any i < j, the center of Kj lies on the boundary of Ki. Then n = O(3dd)
Monochromatic infinite sets in Minkowski planes
We prove that for any -norm in the plane with and for
every infinite , there exists a two-colouring
of the plane such that no isometric copy of is monochromatic. On
the contrary, we show that for every polygonal norm (that is, the unit ball is
a polygon) in the plane, there exists an infinite such that for every two-colouring of the plane there exists a
monochromatic isometric copy of .Comment: 9 pages, 2 figure