Inequalities for the Volume of the Unit Ball in Rn

Let n n 2 Ž1 n 2.be the volume of the unit ball in Rn. We determine the best possible constants a, b, A, B, , and such that the inequalities a nn Ž1n 1. n b nn Ž1n 1. , ʹŽn A. Ž2 . n 1 n ʹŽn B. Ž2 ., and Ž1 1 n. 2n Ž n 1 n 1. Ž1 1 n. are 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.