Let \(Y\) be a closed oriented 3-manifold with \(b_1(Y)\neq 0\), and suppose that \(Y\) contains no non-separating 2-spheres or tori. For such a \(Y\), the dual Thurston norm can be defined on \(H^2 (Y;\mathbb{R})\) by the formula \[ | \alpha| =\sup_\Sigma \bigl\langle \alpha, [\Sigma] \bigr\rangle/ \bigl(2g (\Sigma)- 2\bigr), \] the supremum being taken over all connected oriented surfaces \(\Sigma\) embedded in \(Y\) whose genus \(g(\Sigma)\) is at least 2 [\textit{W. P. Thurston}, A norm for the homology of 3-manifolds, Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)]. If \(Y\) is irreducible, then the authors prove that the unit ball of the dual Thurston norm on \(H^2(Y; \mathbb{R})\) consists of the classes \(\alpha\) whose \(L^2\) norm is not larger than the \(L^2\) norm of \(s_h/4\pi\) for all metrics \(h\) on \(Y\), where \(s_h\) denotes the scalar curvature.