Is it true that there does not exist a closed convex plane curve containing an infinite number of segments, belonging to distinct lines each?
Unless I am misunderstanding the question, the answer is NO. Consider a semicircle whose diameter is the segment $[0, 1]$ on the $x$-axis. Now, consider the curve composed of the diameter, and the chords joining points on the semicircle with arguments $\pi/n$ to $\pi/(n+1)$ (and their reflection in the $y$-axis). This seems to be an example of the convex curve of the type you claim does not exist.