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Two questions jump into my mind when I was working with cone, I feel they are very related to each other. one I asked here "Is $T(C) \subseteq R^m$ closed?"

Second one is: If $C_1$ and $C_2$ are two closed, convex cone in $R^n$ then is $$C_1 + C_2 = \{c_1 + c_2 \; | \; c_1 \in C_1 , ~ c_2 \in C_2\} $$

a closed set?

The answer for my first question is No, I feel the answer of my second question is No as well but I dont have an explicit example to confirm that.

Red shoes
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1 Answers1

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OK, a counter-example.

Let $C_1$ be the closed convex cone $x^2+y^2\leq z^2$, $z\geq 0$ in $\mathbb{R}^3$, and $C_2$ be the half-line $t(1,0,-1)$, $t\geq 0$.

Then $C_1+C_2=\operatorname{conv}(C_1\cup C_2)$ does not contain $(0,1,0)$, but it contains $$ \overbrace{(-t,1+t^{-1},t + \frac{t^{-1} + 2t^{-2} + t^{-3}}2)}^{\in C_1}+\overbrace{(t,0,-t)}^{\in C_2}=(0,1+t^{-1},\frac{t^{-1} + 2t^{-2} + t^{-3}}2) $$ for all $t\gg 1$.

user10354138
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    I dont agree ! How does the compactness of $(C_1\cap\bar B)+(C_2\cap\bar B)$ imply the closedness of $C_1 + C_2$? – Red shoes Jun 25 '19 at 01:18
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    Take the closed ball centered $(0,1)$ with radius $r=1$ in $R^2$, and think about the cone generated by that. – Red shoes Jun 25 '19 at 03:10
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    Right, wasn't thinking straight. Cooked up a counterexample. – user10354138 Jun 25 '19 at 03:43
  • Nice Example, Thank you. – Red shoes Jun 25 '19 at 04:19
  • Nice counterexample, +1, but how on earth did you come up with that formula for points that get arbitrarily close to $(0,1,0)$? What was the thought process that led you to that expression? – J_P Jun 25 '19 at 15:05
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    @J_P It is the same rotation idea as with the $T(C)$ not closed example that leads to trying $(1,0,-1)$-cone/ray and $(0,1,0)$. At first I use $\sqrt{t^2+(1+1/t)^2}$ for $z$ in the cone, then note it can be approximated. – user10354138 Jun 25 '19 at 15:21
  • Ah, so $(t,1-t^{-1},\sqrt{t^2+1+t^{-2}})$ works and then you get rid of the root. Am I getting this right? – J_P Jun 25 '19 at 15:25
  • yes, that works. It isn't necessary to remove the roots, it just looks nicer :-) – user10354138 Jun 25 '19 at 15:30