I am stuck at the first sentence of the following proof of Jensen’s inequality, appearing in Erhan Çınlar's text "Probability and Stochastics", P70-71:
I am pretty frustrated not only by my being unable to understand it, but also by the fact that the claim is contained in a single sentence, which means usually that the claim is obvious. My failed attempts to show that these expectations form a vector in $D$ were based on the convexity of $D$ (anyway, the sentence begins with "Since $D$ is convex ..."). We know that the expression of points on the line segments looks like a weighted sum: if $x$ and $y$ are in a convex set of $\mathbb{R}^d$, then $\alpha x+(1-\alpha)y, 0\le\alpha\le1,$ is still in it, while by analogy, the expectation is also a kind of weighted average. But I just could not write the vector $\mathbb{E}X$ as a point on the line segment connecting some points in $D$. I am self studying so I don't have instructors or TAs. I'm pretty frustrated, so please lend me a hand in helping me understand the first sentence of the proof. You don't have to follow my attempts; any help will be appreciated. Please note that the reason supporting the sentence should be kind of easy because it's only a single sentence. Thank you.