Consider the square $[0,1]^2$. Assume that this region is divided into $N=K^2$ equispaced grid points. How many convex curves can be drawn in terms of $K$?
The points $(0,1)$ and $(1,0)$ are known to be on the convex curve. I am interested in the answer of this questions due to two reasons:
$1.$ I would like to consider $K\rightarrow \infty$ and compare the total number of convex functions to the total number of other types of functions for example (decreasing, etc.)
$2.$ Later I would like to write a program to realize all such discrete convex functions in order to perform optimization over all these functions (for a specific $K$).
I would be also happy to hear any ideas about how to realize this algorithmically.
Here is an example, where one can see examples of three different convex functions:
In this example there are altogether $N=121$ grid points and the distance between each neighboring pair of grid points both in $x$ and $y$ direction is $0.1$.
So, for every value on the $x$ axis, there will be a corresponding value on the $y$ axis.
ADDED (15.07.2018): I programmed and obtained the grid. Using this approach it is impossible to get all convex functions as $N\rightarrow \infty$. I bet we are not even near. Consider the convex function which linearly decreases from $(0,1)$ to $(0.4,0.2)$ and again linearly decreases from $(0.4,0.2)$ to $(1,0)$. It is impossible to get this function with this approach. No mater how fine the grid is, there are infinitely many other functions which are not achieved.