Let $X\subseteq R^d$ and $u\notin conv(X)$.
I want to prove that any $y\in conv(X\cup u)$ can be written as $\lambda u + (1 − \lambda)x$ for some $x \in conv(X)$ and $λ \in [0,1]$.
I intuitively get it but not sure how to prove it. Can anyone guide me?
According to the Caratheodory's Theorem (https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)), any point $y\in \text{conv}(X\cup u)\subset R^{d}$ can be written as $$ y = \sum_{i = 1}^{d+1}{\lambda_i x_i} $$ where $x_i\in (X\cup u)$ and $\sum_{i = 1}^{d+1}{\lambda_i} = 1$, $\lambda_i \ge 0$. If $x_i\neq u$ for any $i = 1, \cdots, d+1$, then $y\in\text{conv}(X)$.
Now suppose $x_{d+1} = u$. If $\lambda_{d+1} = 1$ then $y = u$, which satisfies your conclusion. If $\lambda_{d+1}\neq 1$, then $\sum_{i = 1}^{d}{(1 - \lambda_{d+1})^{-1}\lambda_i} = 1$. So we have $t = \sum_{i = 1}^{d}{(1 - \lambda_{d+1})^{-1}\lambda_i x_i} \in \text{conv}(X)$, and thus $y = (1 - \lambda_{d+1})t + \lambda_{d+1}u$, which completes the proof.
The key point of this proof is to notice that $y$ can be written as a convex combination of finite number of points, and rewrite the combination of $x_i$'s as some $t\in\text{conv}(X)$.