For a random scattering of points, in a bounded area, the goal is to find the largest circle that can be drawn inside those same bounds that does not enclose any points. Solving this problem with a genetic algorithm requires deciding how to encode as a genome, information sufficient to represent any solution. In this case, the only information we need is the center point of the circle, so our genome of point $p_i$ will look like $(x_i, y_i)$, representing the Cartesian coordinates.
In this case, what does each of the genetic operators mean for this simplistic genome? Geometrically speaking, what would a 1-point crossover look like in this case? What about mutation?
This is my answer, but I am not sure.
Consider two individuals with 2 variables each (2 dimensions), $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$. For each variable, the parent who contributes its variable to the offspring is chosen randomly with equal probability. Geometrically speaking, a 1-point crossover would represent a quadrilateral in the Cartesian space, where one of the diagonals is formed by the parents and the other one by the offsprings $c_1=(x_1, y_2)$ and $c_2=(x_2, y_1)$.
On the other hand, a mutation operator is an r-geometric mutation under the metric $d$ if all its offsprings are in the $d$-ball of radius $r$ centered in the parent.
The radius (fitness function) would be the distance between the center (genome) and the closest point (star) from the random points in the bounded area.
