Finite packing and covering by congruent convex domains
For a given convex domain \(K\), let \(H(K)\) be a circumscribed polygon with at most six sides whose area is minimal. According to the celebrated result by L.~Fejes Tóth, if a polygon \(P\) having at most six sides contains \(n\) non-overlapping congruent copies of \(K\), then for its area we have \(A(P)\geq n\cdot A(H(K))\). In this paper, the author provides a stronger version of this result: If a convex domain \(D\) contains \(n\) non-overlapping congruent copies of a convex domain \(K\), then \(A(D)\geq n\cdot A(H(K))\) provided that \(n\geq N\) where \(N\) depends only on \(K\). It is shown that the \(N\) does depend on \(K\) (even for centrally symmetric domains), and cannot be chosen to be an absolute constant. The constant \(A(H(K))\) is not optimal for the typical convex domain \(K\) (in the sense of the Baire category). On the other hand, if \(K\) is centrally symmetric, then one can choose \(H(K)\) to be centrally symmetric according to C.~H.~Dowker. Thus there exists a lattice tiling by translates of \(H(K)\), and hence taking large domains as \(D\) shows that the constant \(A(H(K))\) is optimal in this case. As a corollary, the author obtains the following. For a centrally symmetric convex domain \(K\) that is not a parallelogram, let \(D_n\) be a convex domain with minimal area that contains \(n\) non-overlapping congruent copies of \(K\). Then \(c_1\cdot\sqrt{n} < r(D_n) < R(D_n) < c_2 \cdot\sqrt{n},\) where the positive constants \(c_1\) and \(c_2\) depend on \(K\). In case of coverings some extra assumptions necessary: two convex domains \(C_1\) and \(C_2\) are called non-crossing if there exist complementary half-planes \(I^-\) and \(I^+\) such that \(I^-\cap C_1 \subset C_2\) and \(I^+\cap C_2 \subset C_1\). For a convex domain \(K\), let \(\widetilde H(K)\) be an inscribed hexagon with at most six sides whose area is maximal. If a convex domain \(D\) is covered by \(n\) non-crossing congruent copies of a convex domain \(K\), then \(A(D)\leq n\cdot A(\widetilde H(K)) \) provided that \(n\geq\widetilde N\) where \(\widetilde N\) depends only on \(K\). If \(K\) is centrally symmetric, then one can choose \(\widetilde H(K)\) to be centrally symmetric according to C.~H.~Dowker. Thus there exists a lattice tiling by translates of \(\widetilde H(K)\), and hence taking large domains as \(D\) shows that the constant \(A(\widetilde H(K))\) is optimal. On the other hand, the constant \(A(\widetilde H(K))\) is not optimal for the typical convex domain \(K\).
covering by non-crossing congruent domains
scientific article; zbMATH DE number 2051388