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?

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.