I want to prove the statement:
$f$ is convex on $\Bbb{R}^n$ $\quad\Leftrightarrow\quad$ $\forall x\in\Bbb{R}^n,y\in \Bbb{R}^n$ and $\theta$ with $0\le\theta\le1$, $\phi(\theta)=f(\theta x+(1-\theta) y)$ is convex.
Is there any simple proof?
I have proved this using convex function's definition, but it is so long.
I think there may exist concise proof!!
Thank you for reading my question.
Let's think of this in terms of ideas rather than formulae. $\phi(\theta)$ is a function that, for fixed $x$ and $y$, goes a certain distance along the line between $x$ and $y$ and checks the value of the original function at that point. When we take $\theta_1$ and $\theta_2$ and draw a line between them and check the value at a point on that line, we can construct an $x'$ and a $y'$ that correspond to the location specified by $\theta_1$ and $\theta_2$ and check the value of $f$ on the corresponding point on that line and get the same number back. Since $f$ is convex, it then follows that $\phi$ is convex.
If this idea is clear, it shouldn't be too hard to just sit down and write the inequality that demonstrates convexity.
