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Give an example of a closed subset $S$ of $\mathbb{R}^{2}$ such that conv $S$ is not closed.

Then $S$ is closed, and conv $S$ is open, because conv $S$ is the upper half-plane.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 4

Hyperplanes

Vectors

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to find an example of a closed convex set in our two such that its profile is not empty. But the convex hole of the profile is not the same as the set. So perhaps the easiest example I can think of such a closed conduct set s is going to be a ray in our two. So we have an end point and then from that a half line, well, question is what is the profile of this set profile? P only contains the end point. So the point where the ray originates from so it follows that P is not empty. But convex hole of P is going to be equal to P since P contains only single endpoint, which, as we know, is not the same as s.

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