Intuition behind convex functions

Question

For me, possibly the most out-of-nowhere definition of the first semester of Calculus was the following definition of a convex function and its equivalents.

Function $f$ is convex on the interval $J$ if for $\forall x,y\in J$ and $\forall\lambda\in (0,1)$ is $$f(\lambda x + (1-\lambda )y)\leq\lambda f(x) + (1-\lambda )f(y)\tag1$$

Equivalently if $\forall u,v,w\in J:u<v<w$ $$f(v)(w-u)\leq f(w)(v-u)+f(u)(w-v)\tag2$$

or

$$\frac{f(v)-f(u)}{v-u}\leq \frac{f(w)-f(v)}{w-v}\tag3$$

I'm looking for an intuition, or visual representation of what these three definitions "actually" mean.

(3), being very similar to the definition of a derivative, is the only one that makes sense to me, that is: a function is convex if the slope between points $(u,f(u))$ and $(v,f(v))$ is lesser than the slope between $(v,f(v))$ and $(w,f(w))$.

(2) seems to look at areas of rectangles, however, that is about everything I could say about it.

(1) Got it! $f(\lambda x + (1-\lambda )y)$ is the functional value of a point between $x$ and $y$ and $\lambda f(x) + (1-\lambda )f(y)$ is a point between $f(x)$ and $f(y)$ on a slope between the two points, thus represented by the fact that the slope is never below the functional value. I can now see that it represents the fact that the slope between $x$ and $y$ is always above the function, I don't see, however, how $\lambda f(x) + (1-\lambda )f(y)$ is a point on the slope.

Thanks for any help!

Answer 1

I have a little time now, I'll try to elaborate more later:

(1) Is very well visualised on this picture from wikipedia. $\lambda x + (1-\lambda )y$ is a value that for $\lambda\in (0,1)$ always is between $x$ and $y$ and so $f(\lambda x + (1-\lambda )y)$ is the functional value of a point between $x$ and $y$ and $\lambda f(x) + (1-\lambda )f(y)$ is a point between $f(x)$ and $f(y)$ on a line between the two points, thus representing the fact that the line joining any two points on the function is never below the functional value.

(3) Can be best understood through derivatives. It is an inequality of two slopes , saying that a function is convex if the slope between points $(u,f(u))$ and $(v,f(v))$ is lesser than the slope between $(v,f(v))$ and $(w,f(w))$.

(2) As JavaMan pointed out, "the inequalities (2) and (3) are exactly equivalent", "In other words, they should have the same intuitive meaning"

Further comment on the intuition of convexity was given by Raskolnikov, who said

Replace the inequality by an equality and you get a linear function. Thus, you could say that convexity is a form of "sublinearity"

Thanks to all for help.