Let $\Omega \subset \Bbb R^n$ be a convex domain and $f: \Omega \to \Bbb R $ and $f \in \mathcal C^2(\Omega)$.
Let $x_B $ be the barycentre of $\Omega$ with $$x_B:= \frac{\int_\Omega x \,dx}{\int_\Omega 1\,dx}$$
Show that $$\int_\Omega f(x) \,dx=f(x_B) \int_\Omega1\,dx+ \mathcal O\left(\int_\Omega1\,dx \cdot \sup_{x,y\in\Omega}\|x-y\|_2^2\right)$$ if $\sup_{x,y\in\Omega}\|x-y\|_2 \to0$
I've got a lack of knowledge in (functional) analysis of 2nd, 3rd and 4th term and wanted to know whether there are important sentences,etc. which I don't know and that I should know to solve this problem.
For instance I've read in a book, that a convex domain means that $\Omega$ is an open connected non-empty subset and it's convex that means that $\{(\lambda+1)x+\lambda y\mid 0 \le \lambda \le 1, x,y \in\Omega\}$, especially it's path-connected and it's also star-shaped and contractible. And I'm not sure how the integral over $\Omega$ looks like. Maybe someone could name a book and important topics which I should read through.
Thanks in advance.