Clarkson and Random Clarkson Inequalities for Lr(X)

We first show how (p,p′) Clarkson inequality for a Banach space X is inherited by Lebesgue-Bochner spaces L_r(X), which extends Clarkson's procedure deriving his inequalities for L_p from their scalar versions. Fairly many previous and new results on Clarkson's inequalities, and also those on Rademacher type and cotype at the same time (by a recent result of the authors), are obtained as immediate consequences. Secondly we show that if the (p, p') Clarkson inequality holds in X, then random Clarkson inequalities hold in L_r(X) for any 1 ≤ r ≤ ∞; the converse is true if r = p'. As corollaries the original Clarkson and random Clarkson inequalities for L_p are both directly derived from the parallelogram law for scalars.