Convex functions

A function \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) is convex if and only if the function \(g:\mathbb{R}\rightarrow \mathbb{R}\) given by \(g(t)=f(x+ty)\) is convex (as a univariate function) for all \(x\) in the domain of \(f\) and all \(y∈\mathbb{R}^{n}\). (The domain of \(g\) here is all \(t\) for which \(x+ty\) is in the domain of \(f\)).

• The theorem simplifies many basic proofs in convex analysis but it does not usually make verification of convexity that much easier as the condition needs to hold for all lines (and we have infinitely many).

• Many algorithms for convex optimization iteratively minimize the function over lines. The statement above ensures that each subproblem is also convex optimization problem.