I have been reading about star shaped sets and support cones from this article. I am wondering about the definition of the cone as to why is it defined this way? Why is it $C-a$? I am familiar with the definition given here. And what exactly is the apex of a cone and can you give an example of a cone whose apex does not belong to the cone.
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$\begingroup$ $C-a$ means the set $C-a := \{c-a\colon c\in C\}$ (this is like Minkowski sum notation). According to this definition, an example of a cone would be the open positive orthant $\{(x,y)\colon x, y >0\}$ in $\mathbb{R}^2$, which has apex the origin $(0,0)$, but the origin is not in the cone. $\endgroup$– Sam HopkinsCommented Apr 3, 2022 at 17:16
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$\begingroup$ @SamHopkins Can you give an example of a cone with more than one apex? $\endgroup$– user332905Commented Apr 3, 2022 at 17:20
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$\begingroup$ In $\mathbb{R}^2$, the half-plane $\{(x,0)\colon x \geq 0\}$ has all of the line $x=0$ as apices. $\endgroup$– Sam HopkinsCommented Apr 3, 2022 at 17:21
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$\begingroup$ Ok. Thank you.. $\endgroup$– user332905Commented Apr 3, 2022 at 17:22
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$\begingroup$ Sorry, I should have written "the half-plane $\{(x,y)\colon x \geq 0\}$" in the previous comment, of course. $\endgroup$– Sam HopkinsCommented Apr 4, 2022 at 12:38
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