A family \(F= \{K_1,\dots, K_k\}\) of convex bodies \(K_i\) in \(\mathbb{R}^d\) is well separated if for every choice of points \(p_i\in K_i\) the set of \(\{p_i\}^k_1\) is affinely independent. A halfspace \(H^+\) is a transversal of \(F\) if its bounding hyperplane \(H\) intersects all members of \(F\); it is an \(\alpha\)-section of \(F\) if \(\text{Vol}_d(K_i\cap H^+)= \alpha\cdot\text{Vol}_d(K_i)\), where \(\alpha= (\alpha_1,\dots, \alpha_k)\) and \(0\leq\alpha_i\leq 1\), for all \(i\) \((1\leq i\leq k)\). The main result is that for any \(\alpha\) and any well separated family of strictly convex bodies the space of all \(\alpha\)-sections of \(F\) is diffeomorphic to the sphere \(S^{d-k}\).