ThmDex – An index of mathematical definitions, results, and conjectures.
Turning a finite number of unsigned real numbers into convex combination coefficients
Formulation 0
Let $\lambda_1, \dots, \lambda_N \in [0, \infty)$ each be an D4767: Unsigned real number such that
(i) \begin{equation} \sum_{n = 1}^N \lambda_n \neq 0 \end{equation}
Then
(1) \begin{equation} \frac{\lambda_1}{\sum_{n = 1}^N \lambda_n}, \dots, \frac{\lambda_N}{\sum_{n = 1}^N \lambda_n} \in [0, 1] \end{equation}
(2) \begin{equation} \sum_{n = 1}^N \frac{\lambda_n}{\sum_{m = 1}^N \lambda_m} = 1 \end{equation}