An inequality for \(L_ p\)-norms with respect to the multivariate normal distribution (Q795402)

Let X be an \({\mathbb{R}}^ k\)-valued random variable with expectation 0 and with a k-variate normal distribution, \(\| \cdot \|\) a norm in \({\mathbb{R}}^ k\), and \(Y=\| X\|\). The author shows the existence of a constant \(c_ p\) such that \(\| Y\|_ p\leq c_ p\| Y\|_ 1\). His constant is \(c_ p=2((p+1)/2)^{1/p}\pi^{(p-1)/2p}, (1<p<\infty)\).