Covering problem of Rado

Problems analogous to Tibor Radó's conjecture but involving other shapes were considered by Richard Rado starting in late 1940s.

A typical setting is a finite family of convex figures in the Euclidean space Rd that are homothetic to a given X, for example, a square as in the original question, a disk, or a d-dimensional cube.

Although the exact value of F(X) is not known for any two-dimensional convex X, much work was devoted to establishing upper and lower bounds in various classes of shapes.

By considering only families consisting of sets that are parallel and congruent to X, one similarly defines f(X), which turned out to be much easier to study.

In 2008, Sergey Bereg, Adrian Dumitrescu, and Minghui Jiang established new bounds for various F(X) and f(X) that improve upon earlier results of R. Rado and V. A. Zalgaller.

Coverings of the unit interval (spread vertically for visual clarity). One can always select a number of segments that are covering this interval such that none of the selected segments overlap each other, and the total length of the segments is greater or equal to 1/2, demonstrated above by the selected (red) segments.
To have the squares be as far apart as possible while still having them overlap, the symbol is used to denote an infinitesimally small scaling value, which expands the square by that amount from the center, and as that value reaches zero, the lower bound is approached.

The total area covered by all squares in each approaches 1 as shrinks, and the amount covered by an optimal selection of squares for these sets approaches 1/4 (where the selected squares are red). Note that there are arrangements which have an optimal selection with total area slightly less than 1/4.