Second-order cone
Definition
The set in 
is a convex cone, called the second-order cone.
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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
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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.
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