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

I saw this theorem online however I can't understand what it really means. I know what convexity is but this is just too confusing, can someone please explain it.

Rodrigo de Azevedo
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Rohaan Manzoor
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  • there is some problem in your wording, the domain of $g$ is supposed to be $\mathbb R$ but $ty\in\mathbb R^n$ and if $x\in\mathbb R$ then $x+ty$ is not well defined. Please review your statement. – zwim Nov 04 '21 at 19:45
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    It is not really saying much. It is just saying that $f$ is convex on some convex domain $D$ iff for any $x,y \in D$ the function $f$ restricted to the segment $[x,y]$ is convex. The definition of convexity (for a function) is essentially a one dimensional thing (for $\lambda \in [0,1]$). – copper.hat Nov 04 '21 at 19:47

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