Given a convex 2D region $C$ and a positive integer $N$. We need to cover $C$ with $N$ rectangles such that the sum of the areas of the $N$ rectangles is the least – no further constraints on the dimensions of the $N$ covering rectangles. Then,
Are we guaranteed to get an 'optimal $N$-rectangle cover' of $C$ if we insist that the orientations (direction of the length) of all $N$ rectangles ought to be the same? (Note: If the answer is "yes", finding algorithms for 'optimal $N$-rectangle cover of $C$' would become easier)
If answer to (1) is "yes", one can ask: for the same $C$, if $N$ is varied, can the orientations of the 'optimal rectangle covers of C' always be chosen for every $N$ from a very small set? One guesses one can choose from only 2 possible orientations and get an optimal rectangle cover of $C$ for any $N$.
What about higher dimensional analogs to this question?
Note: Not sure if the following broader class of problems has been explored...
Covering a given convex region $C$ with a specified number $N$ of mutually similar instances of any specified shape - the 'covering units' could be $N$ circles of possibly varying sizes or $N$ squares of not necessarily same side ... - such that the total area of the covering units is minimum and with no constraint on the sizes of the instances of the covering shape being used.
Guess: In the case of covering $C$ with $N$ rectangles, the best layout (one which minimizes the total area of the $N$ covering units) is always such that the $N$ covering units have no overlaps among themselves. Indeed, if $C$ could be non-convex, the covering units in the best layout may overlap.
Note: This guess is not applicable if the covering unit shape 's' is a circle or a convex polygon with large number of sides. Even if we consider covering a convex shape C with N equilateral triangles, it appears that at least for some C and N, the covering equilateral triangles have to necessarily overlap if their total area is to be least.
Note: Other optimizations such as minimizing the sum of the perimeters of the covering units also could be thought about.