I am writing a paper and at some point I make use of the proposition "Any compact, convex set, may be represented as the set of solutions to a (possibly infinite) set of linear inequalities." Is this a well-known theorem or result so that I can cite some paper or do I need to prove this? Or may be this is trivial enough for the reader to prove for himself.
PS: the community of our readers are not assumed to be linear programming experts, they are physicists.
Your statement holds for any closed convex sets in a Banach space, assuming that by "infinitely many linear inequalities" includes uncountably many such inequalities.
This follows from the well-known fact that a closed convex set $C$ can be written as $$ C = \bigcap\{H\supset C: H \text{ is a closed half-space} \} $$ and the fact that $C\subset H$ can be formulated as a linear inequality. The above representation is a corollary of the Geometric Hahn-Banach theorem. If you're looking for a reference I'd suggest Convex Analysis by Rockafellar or any functional analysis textbook should have this result.