# A sufficient condition for being the boundary of one’s convex hull?

Let $$Asubsetmathbb R^n$$ be such that:

1. every non-zero linear functional is maximized by a unique point of $$A$$

2. every point of $$A$$ is a point where some linear functional achieves its maximum over $$A$$ (i.e., every point of $$A$$ is an exposed point, though normally exposed points are only defined for convex sets)

3. the boundary of $$A$$‘s convex hull is contained in the closure of $$A$$.

Does it follow that $$A$$ is closed, and hence equal to the boundary of its convex hull?

It doesn’t follow in the absence of (3). Let $$A = { e^{itheta}: 0letheta.