Is compactness required for the set in Fenchel-Eggleston theorem Suppose $X$ is a subset of $\mathbb{R}^n$ such that it has $n$ connected components. Then Fenchel-Eggleston theorem (Theorem 18 in [1]) tells that every point in the convex hull of $X$ can be written as a convex combination of at most $n$ points.
Notice that there is no assumption about compactness of $X$ in this theorem.
In the Appendix [2] the authors specify that $X$ needs to be compact for Fenchel-Eggleston theorem to hold.
Can someone please clarify if compactness is required or not?
[1] H. G. Eggleston, Convexity. New York: Cambridge university press, 1958.
[2] H. Witsenhausen and A. Wyner, “A conditional entropy bound for a pair of discrete random variables,” IEEE Trans. Inf. Theory, vol. 21, pp. 493–501, Sep. 1975.

I have found the answer to my question in [1]. I just wanted to post it if anyone is interested in the answer.
The authors specifically mention that Fenchel has shown the result for $X$ is connected and compact. Furthermore,[2] shows that the compactness condition of this theorem is superfluous and "connected" can be replaced by "having at most n components." So essentially we do not need compactness for Fenchel-Eggleston theorem.
[1] O. Hanner and H. Rådström, “A generalization of a theorem of fenchel,” Proceedings of the American Mathematical Society, vol. 2, no. 4, pp. 589–593, 1951.
[2] L. Bunt, “Bijdrage tot de theorie der convexe puntverzamelingen proefschrifft groningen noord- hollandsche uitgevers maatschappij,” 1934.
