A family \({\mathcal S}\) of subsets of \(\mathbb{R}^d\) is called by the authors sundered if, for any way of choosing a point from \(r (\leq d + 1)\) members of \({\mathcal S}\), the chosen \(r\) points are affinely independent. The authors mention that this is equivalent to being \((d - 1)\)-separated as defined by \textit{S. Cappell}, \textit{J. E. Goodman}, \textit{J. Pach}, \textit{R. Pollack}, \textit{M. Sharir} and \textit{R. Wenger} [Adv. Math. 106, No. 2, 198-215 (1994; Zbl 0824.52019)]. Let \({\mathcal S} = \{B_1, \dots, B_d\}\) be a sundered family of convex bodies in \(\mathbb{R}^d\). For any partition \((I,J)\) of the index set \(\{1, \dots, d\}\), an \((I,J)\)-support of \({\mathcal S}\) is a hyperplane \(H\) supporting each member of \({\mathcal S}\) such that one of the \(H\)-hyperspaces contains \(\cup \{B_i : i \in I\}\) and the other \(H\)-halfspace contains \(\cup\{B_j : j \in J\}\). This paper is devoted to a new proof of the following result, asserted by \textit{T. Bisztriczky} [Arch. Math. 54, No. 2, 193-199 (1994; Zbl 0717.52006)] and re-proved by Cappel et al. Theorem. If \(\{B_1,\dots,B_d\}\) is a sundered family of convex bodies in \(\mathbb{R}^d\), \(d \geq 2\), then for each partition \((I,J)\) of \(\{1, \dots, d\}\) there are exactly two \((I,J)\)-supports of the family.