Abstract
Methods for improving upper and lower bounds for various coverings of planar sets are proposed. New bounds for various numbers of partition constituents are presented, and suggestions for the generalization of the presented methods are offered.
Highlights
AND BASIC DEFINITIONSLet F be an arbitrary bounded planar set and n ∈ N
Note that dn(F ) does not change if all sets of the covering are assumed to be convex and closed. This conclusion follows from the fact that the diameter of the closure of the convex hull of an arbitrary set F coincides with the diameter of F
For an arbitrary F, the sequence dn(F ) is nonincreasing, since, in the class of all coverings by n + 1 sets, there is a subclass with Fn+1 = ∅ which coincides with the class of all coverings by n sets
Summary
Let F be an arbitrary bounded planar set and n ∈ N. Define the quantity dn = sup dn(F ), where the supremum is taken over all planar sets F of unit diameter. A set Ω ⊂ R2 is called a universal covering set if any planar set F of unit diameter can be completely covered by Ω (i.e., in the plane, there exists a set Ω0 congruent to Ω such that F ⊂ Ω0).
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