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Inequalities for the Volume of the Unit Ball in \mathbb {R}^{n}, II

Authors:

Horst Alzer

Institution:

1. Morsbacher Str. 10, D-51545, Waldbr?l, Germany

Abstract:

We present several sharp inequalities for the volume of the unit ball in $\mathbb {R}^{n}$ ,

$\Omega_{n} = \frac{{\pi ^{n/2}}}{{\Gamma(n/2 + 1)}},\quad n \in {\mathbb{N}}$

. One of our theorems states that the double-inequality

$\frac{A}{{\sqrt n}} \leq (n + 1)\frac{{\Omega_{n + 1}}}{{\Omega_n }} - n\frac{{\Omega_n}}{{\Omega_{n-1} }} < \frac{B}{{\sqrt n}}$

holds for all n ≥ 2 with the best possible constants

$A = (4 - \pi)\sqrt{2} = 1.2139 . . .\quad{\rm and}\quad B = \frac{1}{2}\sqrt{2\pi} = 1.2533 . . . .$

This refines and complements a result of Klain and Rota.

Keywords:

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