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.

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