4. Prove that a subset of R is convex if and only if it is an interval. 5. Show that the union of two convex sets is not a convex set. 7. Given v ∈ R3 0 and ϵ > 0 then let K(v, ϵ) = {x ∈ R3 : ϵ∥v∥∥x∥ ≤ ⟨v, x⟩}. show that K(v, ϵ) is convex. 13. Consider f a convex function and S a convex set. Try that X∗ = {x∗ : f(x∗) ≤ f(x), ∀x ∈ S} It is convex. Please give the answers step by step as explicit as possible. thank you

Q: ercise 6.67. Let E be a subset of R having at least two points. Pro- owing are equivalent:

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A: See attached file for a step by step explanation.

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Q: This answer is incorrect, could you please attempt it again?

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A: As per guidelines, I can do only first three subparts of the question.

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A: As per guidelines, I can do only first three subparts of the question.

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