Inequalities for the volume of the unit ball in Rn

Let Omega (n) = pi (n/2)/Gamma (1 + n/2) be the volume of the unit bah in R-n. We determine the best possible constants a, b, A, B, alpha, and beta such that the inequalities alpha Omega (n/(n+1))(n+1) less than or equal to Omega (n) less than or equal to b Omega (n/(n+1))(n+1), root (n + A) / (2 pi) less than or equal to Omega (n-1)/Omega (n) less than or equal to root (n + B) / (2 pi), and (1 + 1/n)(alpha) less than or equal to Omega (2)(n)/ (Omega (n-1) Omega (n+1)) less than or equal to (1 + 1/n)(beta) are valid for all integers n greater than or equal to 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. (C) 2000 Academic Press.