The authors give a new definition of the convexity index introduced in a recently published paper by \textit{G. Debreu} and \textit{T. C. Koopmans} [Math. Program. 24, 1-38 (1982; Zbl 0495.90063)]. By means of this definition they give then characterizations of the quasiconvexity of the function s defined on \(X_ 1\times X_ 2\times...\times X_ p\) by \[ s(x_ 1,x_ 2,...,x_ p)=f_ 1(x_ 1)+f_ 2(x_ 2)+...+f_ p(x_ p), \] where, for each \(i\in \{1,2,...,p\}\), \(X_ i\) is a non-empty open convex subset of \(R^{n_ i}\) and \(f_ i\) is a non-constant real- valued function on \(X_ i\).