Eine neue Abschätzung der kritischen Determinante von Sternkörpern
type: Item
instance of: scholarly article
title: Eine neue Abschätzung der kritischen Determinante von Sternkörpern
zbMATH Open document ID: 0070.27505
DOI: 10.1007/BF01295169
published in: Monatshefte für Mathematik
publication date: 1956-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/176981
review text: 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.
Mathematics Subject Classification ID: 11H16
zbMATH DE Number: 3117657
zbMATH Keywords: critical determinant
zbMATH Keywords: star bodies
MaRDI profile type: Publication
cites work: Existence theorems in the geometry of numbers
OpenAlex ID: W1964211548
author: Wolfgang M. Schmidt
description: scientific article; zbMATH DE number 3117657