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Can we say that a line is a cone? That is, if I am given set $S$, containing all the points on a line, then that must be cone because for a cone, we require $x \in S$. Then, $cx$ must also be in the set $c \geq 0$

NasuSama
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thetatheta
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1 Answers1

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There are many similar definitions for cones. One of the definitions is: The set A is a cone iff i) for any $c>=0$ and for any x in A, cx is also in A, and ii) {x,-x} is a subset of A iff x=0.

For your question I can say that the ray with vertex to zero is a cone (by the definition i've written). The line is not a cone, because it has both positive and negative elements.

But if you take the definition with just the first condition, than the line is a cone too.

Emo
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    That's not true. If you take the line $y=x$. It surely is a cone. Any multiple (positive or otherwise) of any vector in the set is definitely in the set. – Nitish Feb 16 '14 at 17:50
  • It depends how you define the cone. There are many similar (but not same) definition about the cone. – Emo Feb 16 '14 at 17:56
  • Not entirely true. The two common definitions of cones are "S is a cone if it is closed under nonnegative scalar multiplication" or the same with positive instead of nonnegative. In either case, $y=x$ will surely be a cone. After all, it is a subspace. – Nitish Feb 16 '14 at 18:00
  • In some litterateurs that definition states for wedges. For example see http://www.ams.org/bookstore/pspdf/gsm-84-prev.pdf – Emo Feb 16 '14 at 18:04