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Covering a specific convex shape S with n copies of another specified convex shape S' (which may be different from S) is well studied - for example, https://erich-friedman.github.io/packing/index.html. He we ask about finding for a specified S and n, an optimal S'.

General Question: Given a specific planar convex region C and an integer n, find that convex region C' such that n copies of C' cover C optimally. Let's call C' a 'covering tile' for C.

Here, 'Optimally' could mean the covering tile C' has (1)least area or (2) least perimeter or (3) least diameter or ... - so, what we have given is a bunch of questions. And those covering tiles could be called 'area-optimal/perimeter-optimal... covering tile of order n'. The answers may not be unique eg: When C is a square and n a perfect square, the area optimal covering tile could be either a square or rectangle.

Special case: In the case where C is a triangle, find that triangle T for which its area-optimal covering tile of order 2 has largest area. Obviously, for such a T, the area-optimal covering with 2 congruent tiles is 'most wasteful'.

Remark 1: In this post: From a given triangle, to cut 2 mutually congruent convex pieces that together 'use' maximum area of the triangle we had asked the question: Which is the triangle T such that when 2 mutually congruent convex pieces with maximum possible area are cut from T, the largest fraction area-wise of T is left over? I don't know how that question could possibly relate to the present one.

Remark 2: For a unit square, and for large enough prime n, the area-optimal and perimeter-optimal covering tiles are quite different - indeed, the area optimal ones are rectangles with length 1 and width 1/n and these are too long to be perimeter optimal covering tiles.

Further Question: Find some C and $n$ such that the perimeter-optimal covering tile(s) is not also a diameter-optimal covering tile.

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    $\begingroup$ This needs focus, to clarify what would be an acceptable answer, and to make such an answer feasible. Asking about area-, diameter-, and perimeter-optimal triangle in the same question is not focused enough. And including a remark which you expect to be not useful is not focused. $\endgroup$
    – user44143
    Dec 19, 2021 at 12:19
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    $\begingroup$ You have 93 questions now and have accepted answers to none of them. $\endgroup$
    – user44143
    Dec 19, 2021 at 12:36
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    $\begingroup$ Thanks. reg your first comment, what I tried to state as 'general question' is in fact a bunch of questions - added a clarification. reg the remark which appears as 'not useful', I only pointed to an earlier question which i thought might be related but was not sure. and reg your second comment, i wasn't aware of the provision to accept answers. i shall revisit earlier questions. $\endgroup$ Dec 19, 2021 at 16:45

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