Translative Packing of a Convex Body by Sequences of Its Homothetic Copies

Every sequence of positive or negative homothetic copies of a planar convex body C whose total area does not exceed 0.175 times the area of C can be translatively packed in C. Let C be a planar convex body with area |C|. Moreover, let (Ci) be a finite or infinite sequence of homothetic copies of C. We say that (Ci) can be translatively packed in C if there exist translations σi such that σiCi are subsets of C and that they have pairwise disjoint interiors. Denote by φ(C) the greatest number such that every sequence of (positive or negative) homothetic copies of C whose total area does not exceed φ(C)|C| can be translatively packed in C. In [2] it is showed that φ(T ) = 9 ≈ 0.222 for any triangle T . Moreover, φ(S) = 0.5 for any square S (see [6]). By considerations presented in [7] or in Section 2.11 of [1] we have φ(C) ≥ 0.125. The aim of the paper is to prove that φ(C) ≥ 0.175 for any convex body C. It is very likely that φ(C) ≥ 9 for any convex body C. We say that a rectangle is of type a × h if one of its sides, of length a, is parallel to the first coordinate axis and the other side has length h. Moreover, let [a1, a2]× [b1, b2] = { (x, y); a1 ≤ x ≤ a2, b1 ≤ y ≤ b2 } . The packing method presented in the proof of Theorem is similar to that from [3]. Lemma 1. Let S be a rectangle of side lengths h1 and h2. Every sequence of squares of sides parallel to the sides of S and of side lengths not greater than λ can be translatively packed in S provided λ ≤ h1 and λ ≤ h2 and the total area of squares in the sequence does not exceed 2 |S|. Lemma 2. Let S be a rectangle of side lengths h1 and h2. Every sequence of squares of sides parallel to the sides of S and of side lengths not greater than λ can be translatively packed in S provided λ < h1 and λ < h2 and the total area of squares in the sequence does not exceed λ2 + (h1 − λ)(h2 − λ). Lemma 3. For each convex body C there exist homothetic rectangles P and R such that P is inscribed in C, R is circumscribed about C and that 2 |R| ≤ |C| ≤ 2|P |. Lemma 1 was proved by Moon and Moser in [6], Lemma 2 by Meir and Moser in [5] and Lemma 3 by Lassak in [4]. 2000 Mathematics Subject Classification: Primary: 52C15.