Below is the proof of Poincaré's inequality for open, convex sets. It is taken from "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs" by Giaquinta and Martinazzi. I believe the way the authors use the Fundamental Theorem of Calculus after they write "Noticing that..." cannot be justified on an arbitrary convex set. What they do amounts to connecting any two points by a sequence of $n$ lines, with the $k$-th line parallel to the $x_k$-axis. I am pretty sure this is only possible in an $n$-dimensional cube. Am I wrong about this and can their proof still be made to work for an arbitrary convex set?
Incidentally, I've been looking for a straightforward proof of this this proposition and this is the best I could find, although the proof does not look solid. Could anyone give me a reference with a working proof? It is important that the exponent of the diameter should be exactly $p$ (or $1$ if you take $p$-th roots) and the constant should only depend on $p$ and $n$ (and not on $\Omega$).
