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Eine neue Abschätzung der kritischen Determinante von Sternkörpern. (German) Zbl 0070.27505

Let \(S\) be a bounded \(n\)-dimensional star-body with the origin as centre. Minkowski stated and 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.
Reviewer: C. A. Rogers

MSC:

11H16 Nonconvex bodies
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References:

[1] E. Hlawka, Math. Zeitschrift49 (Berlin 43/44), 285-312.
[2] K. Ollerenshaw, Proc. Camb. phil. Soc.49 (1952), 184-200.
[3] Im allgem. kann diese Abschätzung noch verbessert werden. Man kann z. B.b 2=7 durchb 2=6 ersetzen, muß dann allerdings die folgendenb j teilweise vergrößern.
[4] C. A. Rogers, Annals of Math.,48 (1947), 994-1002 (998). · Zbl 0036.02701 · doi:10.2307/1969390
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