Consider a collection of discs in a maximally random jammed state (also known as random close packing).
If these discs are perfectly circular and have varying radii from a known distribution, that collection still should have an average packing fraction in two dimensions. Define the 2 dimensional packing fraction as the percentage of space occupied by the collection of discs when in a maximally random jammed state. Is there a continuous distribution that would have the highest average packing fraction in 2 dimensions neglecting end effects?
To simplify the problem, it should be assumed that the discs cannot be of infinite nor infinitesimal area, and are frictionless. If necessary, extrema for the radii can be defined but the minimum should be << than the maximum.
If the scope of the question is beyond what can be answered on this site, are there any distributions that are known to have high packing fractions?