Let there be a finite, 2D, rectangular surface S defined by [0, a] x [0, b] with n holes in it. Let P denote a point on the surface such that the distance D between P and the nearest hole edge is greater than or equal to the distance between any other points and their respective nearest hole edge (labeled D' in the attached image). I am attempting to find a hole configuration (size, shape, and arrangement of holes) which both minimizes D and maximizes the area of S not taken up by holes.
If the above is not possible, I'd like to find a configuration which optimizes both these parameters to the greatest extent possible. Does anyone have any thoughts on any software I might be able to use to do this? Are there any relevant papers on this topic I might be able to read?
Here is a link to an image of the problem setup: https://i.stack.imgur.com/UXkul.jpg
Your figure implies that the holes do not need to be convex.
Make a hole that consists of many long narrow slits parallel to two sides of the rectangle and a single long narrow slit parallel another side, connecting all of the other slits.
By increasing the number of slits and correspondingly decreasing the distances between adjacent slits and between the endmost slits and the edges of the rectangle, while decreasing the width of the slits, you can make the area of this hole as small as you want while also making $D$ as small as you want. (For example, if you double the number of slits but divide the width of each slit by $4,$ the area will be approximately half what it was before and $D$ will be approximately half what it was before.)
You can fit $n-1$ very small holes between two of the slits of the large hole in order to satisfy the requirement to have $n$ holes.