I was following the textbook by Boyd on 'Convex Optimization'. In parallel, I was looking at wikipedia to clear some of my doubts. Then I encountered a conflict between the definitions of cones as well as convex cones w.r.t these two sources.
$\textbf{Wikipedia}$:
A subset C of a vector space V is a cone (or sometimes called a linear cone) if for each x in C and $\textbf{positive}$ scalars $\alpha$, the product αx is in C.
A cone C is a convex cone if $\alpha$x + $\beta$y belongs to C, for any $\textbf{positive}$ scalars α, β, and any x, y in C.
$\textbf{Boyd's book}:$
In this book, the definition uses "non-negative" in contrast to the "positive" (highlighted) usage in wikipedia definition.
Why is this inconsistency? Does not it matter much?
It's a matter of whether the origin is necessarily in the cone.
Per https://en.wikipedia.org/wiki/Convex_cone "Also note that the scalars in the definition are positive meaning that the origin does not have to belong to C. Some authors use a definition that ensures the origin belongs to C"
In the textbook you cite, "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe , the origin is always in the cones considered (e.g., second order cone, semidefinite cone, as well as other cones used in conic optimization). Note that the classic book from 1970, "Convex Analysis" by R. Tyrrell Rockafellar, uses the same convention as the Boyd and Vandenberghe book. I don't know why the Wikipedia article uses a different convention.