Random Clarkson inequalities and LP version of Grothendieck' s inequality

Abstract

In a recent paper Kato [3] used the Littlewood matrices to generalise Clarkson's inequalities. Our first aim is to indicate how Kato's result can be deduced from a neglected version of the Hausdorff-Young inequality which was proved by Wells and Williams [11]. We next establish "random Clarkson inequalities".. These show that the expected behaviour of matrices whose coefficients are random ±1's is, as one might expect, the same as the behaviour that Kato observed in the Littlewood matrices. Finally we show how sharp LP versions of Grothendieck's inequality can be obtained by combining a Kato-like result with a theorem of Bennett [1]on Schur multipliers.