A proof of the convexity of the range of a nonatomic vector measure using linear inequalities (Q1322879)

The paper provides a proof to the convexity part of A. A. Liapunov's theorem, that asserts that the range of an atomless finite dimensional vector measure is convex and compact. The present proof follows, e.g., the approach of \textit{P. R. Halmos} [Bull. Am. Math. Soc. 54, 416-421 (1948; Zbl 0033.052)] reducing however a key step that Halmos proves directly to a result on existence of extreme solutions of linear inequalities.