Author’s abstract: Let \(B\) be a Banach space and \(X\subset B\) a normal cone such that the norm is monotone on \(X\) for the order determined by \(X\). We study the sup, denoted by \(i(X)\), of the \(q\geq 1\) such that, for each \(\varepsilon>0\) and each \(n\), there are \(x_ 1,\dots, x_ n\) in \(X\) such that: \[ (1-\varepsilon) \| (a_ k)\|_ q\leq \Biggl\| \sum_ 1^ n a_ k x_ k \Biggr\| \leq (1+\varepsilon) \| (a_ k)\|_ q, \] for all \(a_ 1,\dots, a_ n \geq 0\), where \(\|\;\|_ q\) is the norm in \(\ell^ q\). We prove that \(i(X)\) is the inf of the \(p\) for which we have: \[ \Biggl( \sum_ 1^ n \| x_ k\|^ p \Biggr)^{1/p}\leq S_ p \Biggl\| \sum_ 1^ n x_ k \Biggr\| \qquad \text{for all } x_ 1,\dots, x_ n \text{ in } X. \] The proof uses a similar theorem of J. L. Krivine [Ann. Math., II. Ser. 104, 1-29 (1976; Zbl 0329.46008)] 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 of factorization of operators with values in a \(L^ p\)-space, to the case of normal cones, contained in a Banach space.
[For part I see C. R. Acad. Sci., Paris, Sér. I 314, No. 7, 535-539 (1992; Zbl 0765.46009)].