Summary: Let \(m\) and \(n\) be integers with \(0 0\}= \dim_H\mu- m \] provided that \(\dim_H\mu> m\). Here \(\mu_{V,a}\) is the sliced measure and \(V^{\perp}\) is the orthogonal complement of \(V\). If the \((m+d)\)-energy of the measure \(\mu\) is finite for some \(d> 0\), then for almost all \((n- m)\)-dimensional linear subspaces \(V\) we have \[ \text{ess inf}\{\dim_P \mu_{V,a}: a\in V^{\perp}\text{ with }\mu_{V,a}(\mathbb{R}^n)> 0\}= d_\mu. \] Here \(\dim_P\) is the packing dimension and \(d_\mu\) is a constant defined by means of the convolution of \(\mu\) with a certain kernel. We also deduce corresponding results for the upper packing and upper Hausdorff dimensions.