Convex Holes

Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon).

As an example, the image below shows a set of twenty points and a few such convex holes.
The convex hole shown as a red heptagon has an area equal to $1049694.5$ square units, which is the highest possible area for a convex hole on the given set of points.

For our example, we used the first $20$ points $(T_{2k – 1}, T_{2k})$, for $k = 1,2,\dots,20$, produced with the pseudo-random number generator:

\begin{align}
S_0 &= 290797\\
S_{n + 1} &= S_n^2 \bmod 50515093\\
T_n &= (S_n \bmod 2000) – 1000
\end{align}

i.e. $(527, 144), (-488, 732), (-454, -947), \dots$

What is the maximum area for a convex hole on the set containing the first $500$ points in the pseudo-random sequence? Specify your answer including one digit after the decimal point.

This problem requires computational algorithms and knowledge of computational geometry. It has been discussed and often solved in computer science and competitive programming contexts.

Firstly, the given problem states, that we are operating on a plane with a set of points produced by a pseudo-random number generator.

We know:
$s_{0} = 290797$
$s_{n+1} = s_{n}^2 \;mod\; 50515093$
$t_{n} = s_{n} \;mod\; 2000 – 1000$

Our aim is to find maximum area of a convex hole on a set containing first 500 points in the sequence.

This problem can be solved by using the Graham scan algorithm, which is a method of computing the convex hull of a finite set of points in the plane.

The steps of the Graham scan algorithm are:

1. Find the point with the lowest y-coordinate, break ties by choosing the point with the lowest x-coordinate. This point is the start point s.

2. Sort the remaining points based on the angle that each point makes with the start point and the positive x-axis. If two points make the same angle, remove the one that is closer to the start point.

3. Move the points in the sorted array to a stack, as follows:
– Push the first three points in the sorted array to the stack.
– For each remaining point in the sorted array, while the top two points on the stack and the current point do not make a counter-clockwise turn, pop the top point from the stack.
– Push the current point to the stack.

4. The points remaining on the stack are the vertices of the convex hull in counterclockwise order, beginning with the point with the smallest y-coordinate.

Then, to find the maximum area of a convex hole, you can use the formula for the area of a convex polygon.

Note: These steps require programming skills to implement. I highly recommend learning Python or C++ if you want to dive deep into this type of problem and solve it yourself.

Since the problem doesn’t provide specific points we could use as an example, you will need to apply the algorithm and calculate the values yourself given the 500 pseudo-random points.

Remember that because of the nature of iterative approximation algorithms, the result may depend on the number of iterations performed and the specifics of the algorithm itself.

The final specific answer would be obtained by writing a program to generate the points, calculate the convex hull using Graham’s scan algorithm and then calculate the maximum area of the convex polygon formed.

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