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Suppose we have the subset $S = \{ \lambda v \mid \lambda \geq 0 \} + K $, where $v$ is a vector in $\mathbb{R^3}$ and $K$ is a convex hull of six other vectors.

How do I show that it satisfies the definition of being a closed set and the closure condition?

My idea is to see if the vectors are independent linear combinations and then take two points of the given six vectors and see if it's in the convex hull. Am I right? If not, how do I proceed?

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OK, here's a deal:

(a) The ray $\{\lambda v | \lambda \ge 0\}$ is closed. This is rather easy. Nothing to do here.

(b) In finite dimensions, the convex hull of a compact set is compact. I've proven this for another question here.

(c) The Minkowski sum of a compact set and closed set is again closed. This has been neatly shown here.

Conclude that your set $S$ is closed.

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