Inequalities and sphere-packing in \(\ell _ p\)
Consider the classical sequence spaces \(\ell^ r\), \(1\leq p<\infty\); least upper bounds for the set of real numbers \(r_ p\) such that infinitely many balls of radius \(r_ p\) can be packed in the unit ball of \(\ell^ p\) are known. For \(\rho >r_ p\) it is interesting to know the maximum (finite) number of balls of radius \(\rho\) which can be packed in \(\ell^ p:\) some estimates go back to \textit{J. A. C. Burlak}, \textit{R. A. Rankin} and \textit{P. A. Robertson}, Proc. Glasgow Math. Assoc. 4, 22-25 (1958; Zbl 0087.108). By using nice inequalities, sharp estimates concerning the above numbers are given in the present paper, both for the case \(1\leq p\leq 2\) and for \(2\leq p<\infty\). Also, by combining Theorem 10 here with some results by \textit{D. Amir} [Pac. J. Math. 118, 1-15 (1985; Zbl 0529.46011)] the value of the Jung constant in \(\ell^ p\) (p\(\neq 2)\) is obtained.
scientific article; zbMATH DE number 4046390