If $P$ is a set of vectors $\textbf{x}_i$'s where every $\textbf{x}_i$ is of dimension $d$ and $|P|=K$. In this case at many places I have seen that the vectors $\textbf{x}_2-\textbf{x}_1,\textbf{x}_3-\textbf{x}_1\cdots,\textbf{x}_K-\textbf{x}_1$ are linearly dependent. This is a very crucial point in understanding the proof of Caratheodary theorem but somehow I do not understand how this (linearly dependent part) is true. I will be very thankful to you for shedding some light over it.
Here $K$ is greater than $d+1$
I refer to the wikipedia page of "Caratheodory's Theorem (convex hull)". They assume that $K>d+1$.
$$\left| \left\{ x_i-x_1|i=2,\ldots,K\right\}\right|>d$$
If there are more than $d$ vectors in $\mathbb{R}^d$, they cannot be linearly independent.