Clarkson type inequalities and their relations to the concepts of type and cotype

We prove some multi-dimensional Clarkson type inequalities for Banach spaces. The exact relations between such inequalities and the concepts of type and cotype are shown, which gives a conclusion in an extended setting to M. Milman’s observation on Clarkson’s inequalities and type. Asimilar investigation concerning the close connection between random Clarkson inequality and the corresponding concepts of type and cotype is also included. The obtained results complement, unify and generalize several classical and some recent results of this type.