Return to Question

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, 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 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.

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.

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.

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, 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 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.

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.

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.