Return to Revisions

By Jensen's inequality, for $p\ge2$, $$\Big(\int_{[0,1)^d}\|x\|_2^p\,dx\Big)^{1/p} \ge\Big(\int_{[0,1)^d}\|x\|_2^2\,dx\Big)^{1/2} =\sqrt{d/3}.$$ So, the trivial upper bound $\sqrt d$ on $\big(\int_{[0,1)^d}\|x\|_2^p\,dx\big)^{1/p}$ is optimal up to a universal constant factor.