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- W1497201547 abstract "The Hadwiger number H(K) of a d-dimensional convex body K is the maximum number of mutually nonoverlapping translates of K that can touch K. We define H∗(K) analogously, with the restriction that all touching translates of K are pairwise disjoint. In this paper, we verify a conjecture of Zong [1997] by showing that for any d1, d2 ≥ 3 there exist convex bodies K1 and K2 such that Ki is di-dimensional, i = 1, 2, and H(K1 ×K2) > (H(K1)+1)(H(K2)+1)− 1 holds, where K1 ×K2 denotes the direct product of K1 and K2. To obtain the inequality, we prove that if K is the direct product of n convex discs in the plane and there are exactly k parallelograms among its factors, then H(K) = 4(4·6+1)/5. Based on this formula, we also establish that for every d ≥ 3 there exists a strictly convex d-dimensional body K fulfilling H(K) ≥ 16 35 ( √ 7)." @default.
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- W1497201547 date "2007-01-01" @default.
- W1497201547 modified "2023-09-28" @default.
- W1497201547 title "On Hadwiger Numbers of Direct Products of Convex Bodies" @default.
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