Inequalities for the gamma function and estimates for the volume of sections of B-p(n)

Let B-p(n)= {(x(i)) is an element of R-n; Sigma (n)(1) \x(i)\(p) less than or equal to 1} and let E be a k-dimensional subspace of R-n. We prove that \E boolean AND B-p(n) \(1/k)(k) greater than or equal to \B-p(n)\(1/n)(n), for 1 less than or equal to k less than or equal to (n - 1)/2 and k = n - 1 whenever 1 < p < 2. We also consider 0 < p < 1 and other related cases. We obtain sharp inequalities involving Gamma function in order to get these r esults.