Some questions related to the norm-to-weak upper semi-continuity of the pre-duality map are studied. Let us give the necessary notation and definitions. Denote by \(S_X\) the unit sphere of a Banach space \(X\) and by \(NA(X)\) the set of elements in \(S_{X^*}\) which attain their norm. Given \(x^*\in NA(X)\), the non-empty set \(S_{x^*}= \{x\in S_{X} \colon \langle x,x^*\rangle=1\}\) is its pre-state space, and \(S^{x^*}= \{x^{**}\in S_{X^{**}}\colon \langle x^{**},x^*\rangle =1\}\) its state space. The pre-duality mapping \(\rho\colon S_{X^*}\rightarrow 2^X\), given by \(\rho(x^*)=S_{x^*}\) for all \(x^*\in NA(X)\) and by \(\rho(x^*)=\emptyset\) otherwise, is norm-weak upper semicontinuous ((n-\(w\))usc, in short) at \(x^*\in NA(X)\) if given any weak neighborhood \(V\) of 0 in \(X\) there exists a \(\delta>0\) such that for all \(y^*\in S_{X^*}\) with \(\| y^*-x^*\| <\delta\) we have \(\rho(y^*)\subset\rho(x^*)+V\). An interesting result proved by G. Godefroy and V. Indumathi [Set–Valued Anal. 10, No. 4, 317–330 (2002; Zbl 1042.46008)] is that \(\rho\) is (\(n\)-\(w\))usc at \(x^*\in S_{X^*}\) if and only if \(\overline{S_{x^*}}^{w^*}=S^{x^*}\).
The aim of this paper is to discuss whether the norm-weak upper semi-continuity of the pre-duality map passes to higher odd duals. The authors begin by presenting a simple example showing that the points of (n-\(w\))usc of the preduality map do not in general remain points of (n-\(w\))usc for the preduality map of the higher duals. Next, they give geometric conditions on \(X\) that (n-\(w\))usc points of \(X^*\) continue to be (n-\(w\))usc in all the higher odd duals. One example of such a geometric property is the \(M\)-embeddedness. The authors also study (n-\(w\))usc points in direct sums of spaces. Finally, it is shown that the identity mapping is always a point of (n-\(w\))usc for the dual space \(L(X^*)\) with respect to its predual \(X\otimes_{\pi} X^*\). This latter result is a consequence of the Godefroy–Indumaty characterization of (n-\(w\))usc and classical results about numerical ranges of operators [see F. F.Bonsall’s and J. Duncan, “Numerical ranges of operators on normed spaces and of elements of normed algebras” (Cambridge University Press) (1971; Zbl 0207.44802) for background].