On a conjecture of L.\xa0Fejes\xa0T\xf3th and J.\xa0Moln\xe1r about circle coverings of the plane

Dietmar Dorninger

doi:10.1007/s10998-018-0254-z

Abstract

Tóth and Molnár (Math Nachr 18:235–243, 1958) formulated the conjecture that for a given homogeneity q the thinnest covering of the Euclidean plane by arbitrary circles is greater or equal a function S(q). Florian (Rend Semin Mat Univ Padova 31:77–86, 1961) proved that if the covering consists of only two kinds of circles then the conjecture is true supposed that S(q)le S(1/q) what can be easily verified by a computer. In this paper we consider the general case of circles with arbitrary radii from an interval of the reals. We set up two further functions M_0(q) and M_1(q) and prove that the conjecture is true if S(q) is less than or equal to S(1 / q), M_0(q) and M_1(q). As in the case of two kinds of circles this can be readily confirmed by computer calculations. (For qge 0.6 we even do not need the function M_1(q) for computer aided comparisons.) Moreover, we obtain Florian’s result in a shorter different way.

Highlights

IntroductionWe prove that for given q and β δ(q, β, r1, γ ) has a unique stationary point at which δ(q, β, r1, γ ) assumes its minimum δ0(q, β)
A circle covering K(q) of homogeneity q of the Euclidean plane is a countable set of closed circular discs Ci with radii ri such that every point of the plane belongs to at least one circle of K(q) and q := inf(ri /r j ), i, j = 1, 2
Given q with 0 ≤ q ≤ 1 it is of interest to determine the density D(q) of a thinnest covering of the plane. (For a formal definition of a thinnest covering see, for example, the classical book [8].)

Summary

Introduction

We prove that for given q and β δ(q, β, r1, γ ) has a unique stationary point at which δ(q, β, r1, γ ) assumes its minimum δ0(q, β). This minimum is not always assumed within the limits of the domain B defined by all boundary conditions. Showing by computer aid that M1(q) > S(q) for 0 < q < 0.6 we obtain that S(q) is the smallest of the four functions M0(q), M1(q), S(1/q) and S(q) for any homogeneity q, this way confirming the conjecture of Tóth and Molnár

Dorninger

Two radii in a line

Coverings with only two kind of circles

Conclusions