Covering of the plane by discs (Q1076980)

We say that a compact subset of the Euclidean plane with nonempty interior is a disc if it is bounded by a closed Jordan curve of finite length. Let a and p be given satisfying \(4\pi \leq p^ 2/a<8\sqrt{3}\) and let \({\mathcal D}(a,p)\) be the class of discs with area not less than a and perimeter not greater than p. What is the thinnest covering of the plane by discs from \({\mathcal D}(a,p)?\) A convex disc obtained from a regular k- gon P by joining each two consecutive vertices of P by congruent circular arcs is called a regular arc-sided k-gon. Under some restrictions concerning the intersection of the convex hulls of any two discs of the covering, the authors solve the problem posed above. The thinnest covering arises from a regular hexagonal tiling of appropriate edge- length by joining equal circular segments to the sides of the tiles, so as to obtain regular arc-sided hexagons of area a and perimeter p. Note that the analogous problem of the densest packing of the plane by discs from \({\mathcal D}(a,p)\) is easily solved with the aid of a result of Heppes.