A version of Abel's lemma: Let \(K\) be a convex subset of \(\mathbf{R}^n\) and suppose we have a sequence \(a_1,\ldots,a_n\in\mathbf{R}^n\) so that \(\sum_{i=1}^N a_i \in K\) for all \(0\le N\le n\). If \(1\ge \phi_1\ge\ldots\ge\phi_n\ge 0\), then \(\sum_{i=1}^n \phi_ia_i \in K\) as well.