Some remarks concerning kissing numbers, blocking numbers and covering numbers
The author considers three functionals of convex bodies which are usually investigated in combinatorial geometry. Let \(K\) be a \(d\)-dimensional convex body. The kissing number \(N(K)\) of \(K\) is the largest number of non-overlapping translates of \(K\) which can be brought into contact with \(K\). The blocking number \(Z(K)\) of \(K\) (which was recently introduced by the author) is the smallest number of non-overlapping translates of \(K\) all of which are in contact with \(K\) and which are sufficient to prevent, any other translate of \(K\) from touching \(K\). The covering number \(H(K)\) of \(K\) is the smallest number of translates of \(\text{int} (K)\), the union of which covers the body \(K\). First, the author shows that \(H(K) \leq Z(K)\) for each centrally symmetric convex body \(K\). By the known relation \(N(K) \leq 3^d - 1\) and \(Z(K) \leq N(K)\), this yields \(H(K) \leq 3^d - 1\) if \(K\) has central symmetry, i.e., known upper bounds on \(H(K)\), obtained by Lewin, Petunin, Rogers, and Erdös for centrally symmetric bodies, are improved. Further on, the following phenomenon is proved: when \(d\) is sufficiently large, there are two \(d\)-dimensional convex bodies \(K_1\) and \(K_2\) such that both \(N(K_1) < N(K_2)\) and \(Z(K_1) > Z(K_2)\) hold simultaneously. Finally, the author gives a conjecture regarding the quantity \(Z(K)\), namely \(2d \leq Z(K) \leq 2^d\).
scientific article; zbMATH DE number 769578