| # Second-order cone ## Definition The set in is a convex cone, called the second-order cone.
| Example: The second-order cone is sometimes called ‘‘ice-cream cone’’. In , it is the set of triples with The blue circle corresponds to the set
## Proof of convexity The fact that is convex can be proven directly from the basic definition of a convex set. Alternatively, we may express as an intersection of half-spaces, as follows. From the Cauchy-Schwartz inequality, we have we have Each one of the sets involved in the intersection is a half-space.