My question is the following: let $K$ be a convex set in $\mathbb{R}^n$ and $x$ an element of the interior of $K$. Can I affirm that there exist $z_1,...,z_n \in K$ linearly independent and $\lambda_1,....,\lambda_n > 0$ such that: \begin{equation} x = \displaystyle \sum_{p=1}^n \lambda_p z_p \quad ??? \end{equation} If this statement is true, is there someone who can give me a proof ? Thank you very much and have a nice day !
Characterization of the interior of a convex set
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linear-algebra
real-analysis