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Given an integer $n$ and a convex closed planar curve $C$ ($C$ could be smooth). We need to put $n$ points on $C$ such that (1) the area of the convex hull of these points is maximum. (2) perimeter of the convex hull is maximum.

Question: Are there $C$'s and $n$'s such that none of the possibly many answers (point distributions that maximize area) to question 1 is an answer to question 2 - in other words, if the sets of answers to the two questions are disjoint?

Further question: In 3D, consider putting $n$ points on a closed convex surface such that the convex hull of these $n$ points has (1) max volume, (2) max surface area (3) max edge length. What can one say about the 3 answers - are there cases where all 3 questions/any pair of the questions have different answers?

Note: Maximal tetrahedra inscribed in ellipsoid shows the result that there are many triangles of max perimeter inscribed in a given ellipse but it is not clear to me if the inscribed triangle(s) of max area can be NOT one of these triangles for some ellipses and n's.

One more Question: As has been answered in comments below with example, "there is no reason to expect the maxima(1) and (2) coincide". Then, one wonders which is that C for which maxima of (1) and maxima(2) are 'farthest apart'. Iow, which convex C maximizes the ratios: (1) between the area of the max area inscribed triangle and area of the max perimeter inscribed triangle and (2) between perimeter of max perimeter inscribed triangle and perimeter of the max area inscribed triangle? The C's maximizing these two ratios may not be the same. And this question can be asked for n-gons instead of triangles.

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    $\begingroup$ Do you even know if the case where $C$ is a polygon and $n=3$ is settled? If not, you could trying algorithmically searching for a counterexample, e.g., using the algorithm in the paper Maximum-Area Triangle in a Convex Polygon, Revisited. $\endgroup$
    – Timothy Chow
    Commented Mar 26, 2021 at 15:52
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    $\begingroup$ In general there's no reason to expect that the maxima (1) and (2) coincide. Ellipses (and ellipsoids in higehr dimension) are very special cases because they have continuous families of affine automorphisms, so any maximal-area polygon moves in a continuous family. For typical $C$ there are no nontrivial automorphisms at all. $\endgroup$
    – Noam D. Elkies
    Commented Mar 26, 2021 at 16:35
  • $\begingroup$ Thank you! The paper by Keikha et al give an easily checkable example (the 9-gon shown in fig 3, page 5) - and it has quite different max area and max perimeter inscribed triangles. And from mathoverflow.net/questions/78165/…, there seem to be many convex closed C's each with infinitely many max area and max perimeter inscribed n-gons; are there any C among them such that answer sets to questions 1 and 2 are disjoint (in 3D are there any closed surfaces with the answer sets to questions 1, 2, 3 disjoint)? $\endgroup$
    – Nandakumar R
    Commented Mar 27, 2021 at 5:49

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