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Given convex set $C$ and function $f \colon C \to \mathbb R$, I am told that $f$ is convex if and only if

$$\phi(\lambda) = f(\lambda c + (1-\lambda) c') : [0,1] \to \mathbb R$$

for $c, c' \in C$ is convex on $[0,1]$. I am done with the $\Rightarrow$ implication, but I am struggling with the $\Leftarrow$ one. Any hint would be helpful. Thanks!

Rodrigo de Azevedo
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R__
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    I’m not sure what your definition of a convex function is. One definition is that the epigraph of the function is convex. If you assume that definition, then your iff follows immediately. – NicNic8 Sep 11 '21 at 16:33
  • I'm supposed to use the following def: $f(p_1c +p_2 c') \leq p_1f(c) + p_2f(c')$, $c,c'\in C$ and $p_1 + p_2 = 1$. – R__ Sep 11 '21 at 16:37

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