Inequalities for the Volume of the Unit Ball in Rn

Abstract

Let Ωn=πn/2/Γ(1+n/2) be the volume of the unit ball in Rn. We determine the best possible constants a, b, A, B, α, and β such that the inequalitiesaΩn/(n+1)n+1≤Ωn≤bΩn/(n+1)n+1,n+A/2π≤Ωn−1/Ωn≤n+B/2π,and1+1/nα≤Ω2n/Ωn−1Ωn+1≤1+1/nβare valid for all integers n≥1. Our results refine and complement inequalities proved by G. D. Anderson et al., K. H. Borgwardt, and D. A. Klain and G.-C. Rota.