On generalized kissing numbers and blocking numbers (Q2707488)

Let \(C\) be a convex body in \(\mathbb{R}^{n}\) and \(\alpha >0\). The generalized kissing number \(N_{\alpha }(C)\) is the maximal number of non-overlapping translates of \(\alpha C\) such that each touches \(C\). NEWLINENEWLINENEWLINEThe authors prove the following bounds: \(N_{\alpha }(C)\leq ((1+2\alpha)^{n}-1)/\alpha ^{n}\) with equality if and only if \(C\) is a parallelotope and \(\alpha =1/k\) for a positive integer \(k\), and \(N_{\alpha }(C)\geq 2^{cn} 4\) for \(\alpha <\sqrt{2}\), with a positive constant \(c\) depending only on \(\alpha \). As \(\alpha \) tends to zero, \(N_{\alpha }(C)\) is shown to be asymptotically equal to a certain constant (depending on \(C\)) times \( 1/\alpha ^{n-1}\), thereby extending a result of \textit{L. Fejes Tóth} on polytopes [Stud. Sci. Math. Hung. 5, 173-180 (1970; Zbl 0198.55101)]. NEWLINENEWLINENEWLINEThe generalized blocking number \(B_{\alpha }(C)\) is the smallest number of non-overlapping translates of \(\alpha C\) each touching \(C\) such that no additional translate of \(\alpha C\) with these properties can be inserted. For \(n=3\), the asymptotical behaviour of \(B_{\alpha }(C)\) (for \(\alpha \rightarrow 0\)) is shown to be similar to that of \(N_{\alpha }(C)\). In view of technical difficulties, the case \(n>3\) is only settled for polytopes.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00038].