By Károly Böröczky Jr
This ebook presents an in-depth, cutting-edge dialogue of the idea of finite packings and treatments via convex our bodies. It includes numerous new effects and arguments, collects different key facts scattered in regards to the literature, and gives a entire therapy of difficulties whose interaction was once no longer basically understood sooner than this article. preparations of congruent convex our bodies in Euclidean area are coated, and the density of finite packing and protecting via balls in Euclidean, round and hyperbolic areas is taken into account.
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Example text
B¨or¨oczky Jr. [B¨or1994] and Ch. Zong [Zon1995a]. 9. Covering the Maximal Perimeter We show that a compact convex set of maximal perimeter that can be covered by n congruent copies of a given convex domain K is close to some segment for large n. It is equivalent to show that the inradius is small because if s is a diameter of a convex domain D then D ⊂ s + 3r (D) · B 2 . 23), let h be the maximal distance of points of D from s. Then D contains a triangle T such that s is a longest side, and the distance of the opposite vertex from s is h.
For i < j, let li j be a line witnessing that i and j are noncrossing, and let li+j and li−j denote the half planes such that li+j ∩ j contains li+j ∩ i and − li−j ∩ i contains li−j ∩ j , respectively. We may assume that 2 overlaps l12 , and hence there exist a supporting line l to 2 that intersects 2 in a single − point p, where p ∈ int l12 . After possibly renumbering the domains 3 , . . , n , we may assume the following property for some 2 ≤ m ≤ n: If, for i ≥ 2, i contains p, and p has a neighbourhood Ui such that i ∩ Ui ⊂ 2 ∩ Ui then i ≤ m, and i > m otherwise.
This observation reduces various packing problems to the case when K is o-symmetric. Next, we show that if K is an o-symmetric convex domain then any x ∈ ∂ K is the vertex of an inscribed affine regular hexagon. 4. If K is an o-symmetric convex domain and x ∈ ∂ K then there exists a y such that ±x, ±y, ±(y − x) ∈ ∂ K . In addition, the lattice generated by 2x and 2y is a packing lattice for K . Proof. We choose y to be a common boundary point of K and x + K . For the lattice generated by 2x and 2y, it is equivalent to show z ∈ int2K for z ∈ \o.