Let B be a Banach space and X ⊂ B a normal cone such that the norm is monotone on X for the order determinated by X.
We study the sup, denoted by i(X), of the q ≥ 1 such that, for each E> 0 and each n, there are x1, …, xn in X such that:
for all a1, …, an ≥ 0, where ‖ ‖q is the norm in lq.
We prove that i(X) is the inf of the p for which we have:
The proof use a similar theorem of Kirvine, concerning Banach Riesz spaces. Here conical measures are a useful tool. We establish a link with a preceding work in which we adapt the Maurey theory factorisation of operators with values in a LP space, to the case of normal cones, contained in a Banach space.