# Reference request in convex analysis: algebraic closure vs closure of convex set in finite dimension

I'd like to use the following fundamental result (1) in my paper but I don't know a good reference. Could anyone help?

Setup: for a set $$A$$ in a vector space $$V$$, its algebraic closure $$A^{ac}$$ consists of points $$x\in V$$ that there is some $$v\in V$$ so that for any $$\epsilon>0$$, there is some $$t\in [0,\epsilon]$$ such that $$x+tv\in A$$. I believe the following result (1) is true and fundamental:

(1) If $$V$$ is finite-dimensional and $$A$$ is convex, then $$A^{ac}$$ is the same as the closure of $$A$$ in the usual norm topology.

The references (books) on convex analysis in English that I know do not talk about algebraic closure since they either focus on finite-dimensional spaces or normed vector spaces. Does anyone know a suitable reference?

Alternatively, a reference to any of the following results (2)(3) would be good enough. This involves the notion of an algebraically closed set, which is a set $$A$$ such that for any $$x\notin A$$ and any $$v\in V$$, there is some $$\epsilon>0$$ so that $$x+tv \notin A$$ for all $$t\in [0,\epsilon]$$.

(2) If $$V$$ is finite-dimensional, then a convex algebraically closed set $$A$$ is closed in the usual norm topology.

(3) For an algebraically closed convex set $$A$$, any point $$x$$ outside of $$A$$ is strongly separated from $$A$$, that is, there is a linear functional $$f$$ and a number $$c$$ so that $$f(x) for all $$a\in A$$.

It is easy to note that (2) implies (1) since in a finite-dimensional space, the algebraic closure of a convex set is algebraically closed, thus $$A^{ac}$$ is closed and contains the closure $$\bar{A}$$; conversely $$A^{ac}\subset (\bar{A})^{ac}=\bar{A}$$.

(2) follows from (3) since the strong separation provides each point $$x\notin A$$ an open neighborhood (an open half space) disjoint from $$A$$.

In the book, a related notion $$\mathrm{lin}(A)$$ is defined, which is the union of $$A$$ and points linearly accessible from $$A$$. Here a point $$x$$ is linearly accessible from $$A$$ if there is some $$a\in A$$ such that $$[a,x)\subset A$$. It is easy to note that for any $$A$$ convex, the algebraic closure defined in the question coincides with $$\mathrm{lin}(A)$$.
It is commented in the book on Page 9 part D that $$\mathrm{lin}(A)$$ is the (topological) closure of $$A$$ in a finite-dimensional space if $$A$$ is convex. A more general statement is part (c) of the Lemma on Page 59, which states
$$\mathrm{lin}(A)=\bar{A}$$ for any convex set $$A$$ with nonempty interior in a topological vector space.
If $$A$$ lies in a finite-dimensional space, then the statement above applies to the affine hull of $$A$$ (i.e. the smallest affine subspace containing $$A$$), implying statement (1) in the question.