Summary: Let \(m, n \in \mathbb{N}\) and \(p \in (0, \infty)\). For a finite dimensional quasi-normed space \(X = (\mathbb{R}^m, \Vert\cdot\Vert_X)\), let \[ B_p^n(X) = \left\{(x_1, \ldots, x_n) \in \left(\mathbb{R}^m\right)^n : \sum\limits_{i=1}^n \Vert x_i\Vert_X^p \leqslant 1\right\}. \] We show that for every \(p \in (0, 2)\) and \(X\) which admits an isometric embedding into \(L_p\), the function \[ S^{n-1} \ni \theta = (\theta_1, \ldots, \theta_n) \longmapsto \left|B_p^n(X) \cap \left\{(x_1, \ldots, x_n) \in \left(\mathbb{R}^m\right)^n : \sum\limits_{i=1}^n \theta_i x_i = 0 \right\}\right| \] is a Schur convex function of \((\theta_1^2, \ldots, \theta_n^2)\), where \(|\cdot|\) denotes Lebesgue measure. In particular, it is minimized when \(\theta = \left(\frac{1}{\sqrt{n}}, \ldots, \frac{1}{\sqrt{n}}\right)\) and maximized when \(\theta = (1, 0, \ldots, 0)\). This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body \((B_p^n(X))^\circ\) if the unit ball \(B_X\) of \(X\) is in Lewis’ position. Finally, we prove a lower bound for the volume of projections of \(B_\infty^n(X)\), where \(X = (\mathbb{R}^m, \Vert\cdot\Vert_X)\) is an arbitrary quasi-normed space.