Eine neue Abschätzung der kritischen Determinante von Sternkörpern (Q767155)

Let \(S\) be a bounded \(n\)-dimensional star-body with the origin as centre. Minkowski stated and \textit{E. Hlawka} [Math. Z. 49, 285--312 (1943; Zbl 0028.20606)] proved that the ratio \(Q(S)=V(S)/\Delta(S)\) of the volume \(V(S)\) of \(S\) to its critical determinant satisfies \(Q(S)\geq 2\zeta(n)\). The author uses a technique, developed for another purpose [Monatsh. Math. 59, 274--304 (1955; Zbl 0066.29204)], to obtain an improved lower bound for \(Q(S)\) and to show in particular that \[ Q(S)\geq 3(1+2(\tfrac 12)^n)^{-1}\zeta(n), \quad Q(S)\geq \tfrac {10}3 (1+2(\tfrac 12)^n+7(\tfrac 13)^n)^{-1}\zeta(n), \] and that \(Q(S)>3.418\ldots\) for \(n\) sufficiently large. These bounds are rather better than those obtained independently by the reviewer about the same time [Philos. Trans. R. Soc. Lond., Ser. A 248, 225--251 (1955; Zbl 0066.03603)]. More recently both author and reviewer have obtained further improvements.