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In Boyd-Vandenberghe Convex Optimization and had a question about the proper cone. The following is the property of a proper cone:
• K is convex.
• K is closed.
• K is solid, which means it has a nonempty interior.
• K is pointed, which means that it contains no line (or equivalently, x ∈ K, − x ∈ K =⇒ x = 0).
A proper cone K is pointed (which means that it contains no line). Can somebody please elaborate on what "it contains no-line" mean? P.S. I am new here

Art of Juking
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    Definitions are your friends. Start with the definition of a cone. Is it a subset of a Euclidean space? A vector space? An affine space? Context mattters. Now "it contains no line" is an additional qualification. What precisely a line means depends on your context, but it basically means a one-dimensional subset that extends to infinity in both directions. – hardmath Feb 02 '21 at 16:31
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    Thanks. I considered the two rays that make the cone as lines. Didn't consider the negative infinity. – Art of Juking Feb 02 '21 at 17:25
  • "(or equivalently, x ∈ K, − x ∈ K =⇒ x = 0)" sounds like you answered your own question – Calvin Khor Feb 07 '21 at 03:50

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