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This post continues from Cutting convex regions into equal diameter and equal least width pieces - 2 and Cutting convex regions into equal diameter and equal least width pieces - 3

A basic (and to my knowledge, not yet settled) question posed in above discussions was: If a convex planar region C is divided into n convex pieces such as the maximum value of the diameter among the n pieces is to be minimized, will it automatically guarantee that diameter will be equal among the pieces?

Questions:

  1. Is there some quantity Q', a convex C and an integer n>1 such that when C is divided into n pieces such that maximum of Q' among the pieces is minimized, the values of Q' are not same across the pieces?

Note: Q' should be coordinate independent, like area, diameter, perimeter etc.. Also note that we can ask this question with "maximizing the minimum of Q'"

  1. Is there some quantity Q'' with the property: if any C is divided into n convex pieces such that the average value of Q'' among the pieces is to be minimized, that will guarantee the value of Q'' is automatically equal for all the pieces?

Remarks: If we choose Q'' as something like "minimum x coordinate of a piece" then, the partition of C that minimizes average of Q'' will be made by chords starting at the point on C with least x value and the value of Q'' is indeed, equal over pieces. But we are not looking at such coordinate dependent quantities.

Perimeter does not answer question 2. Indeed, if average perimeter is to be minimized over n pieces, the way is to have one big and n-1 utterly small (tending to zero perimeter) pieces.

And if we are looking for quantities for which maximizing the average over n pieces will automatically equalize its values, diameter satisfies it - the maximum value of the average diameter can be obtained by all pieces having the same diameter (equal to diameter of C itself) but with most pieces degenerate.

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  • $\begingroup$ Thanks. made a couple of edits to clarify these. $\endgroup$ Nov 23, 2021 at 18:02
  • $\begingroup$ Thanks for editing! $\endgroup$
    – user44143
    Nov 24, 2021 at 21:49

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