It appears that the max area inscribed n-gon for an ellipse is not unique - if one inscribes a regular n-gon in a circle with radius a (there are infinitely many orientations for this inscribed regular n-gon) and then compresses the circle along one direction into an ellipse with minor axis b, the inscribed regular n-gon gets pressed into an n-gon that seems to be a max area inscribed n-gon for the (a,b) ellipse.

1. Given an integer n and an ellipse with major and minor axes a and b, how does one find and characterize inscribed n-gon(s) of maximum perimeter?

Guess: the max perimeter inscribed n-gons for an ellipse are probably polygonal closed billiard paths (such polygons for n <=6 are shown on https://mathworld.wolfram.com/Billiards.html). If this guess is right, does it generalize to cases where a closed convex curve allows such polygonal billiard paths with m edges? The guess also implies a unique answer for an ellipse and n; that brings up the issue: given any point P on the ellipse boundary, to find the inscribed n-gon that necessarily contains P and has maximum perimeter.

2. When are the circumscribed n-gons of least perimeter and least area for an ellipse identical?

Note: one can also ask about inscribed and circumscribed n-gons in curves of constant width.

It appears that the max area inscribed $n$-gon for an ellipse is not unique - if one inscribes a regular $n$-gon in a circle with radius a (there are infinitely many orientations for this inscribed regular $n$-gon) and then compresses the circle along one direction into an ellipse with minor axis $b$, the inscribed regular $n$-gon gets pressed into an $n$-gon that seems to be a max area inscribed $n$-gon for the $(a,b)$ ellipse.

1. Given an integer $n$ and an ellipse with major and minor axes $a$ and $b$, how does one find and characterize inscribed $n$-gon(s) of maximum perimeter?

Guess: the max perimeter inscribed $n$-gons for an ellipse are probably polygonal closed billiard paths (such polygons for $n \ge 6$ are shown here). If this guess is right, does it generalize to cases where a closed convex curve allows such polygonal billiard paths with m edges? The guess also implies a unique answer for an ellipse and $n$; that brings up the issue: given any point $P$ on the ellipse boundary, to find the inscribed $n$-gon that necessarily contains $P$ and has maximum perimeter.

2. When are the circumscribed $n$-gons of least perimeter and least area for an ellipse identical?

Note: one can also ask about inscribed and circumscribed $n$-gons in curves of constant width.

It appears that the max area inscribed n-gon for an ellipse is not unique - if one inscribes a regular n-gon in a circle with radius a (there are infinitely many orientations for this inscribed regular n-gon) and then compresses the circle along one direction into an ellipse with minor axis b, the inscribed regular n-gon gets pressed into an n-gon that seems to be a max area inscribed n-gon for the (a,b) ellipse.

1. Given an integer n and an ellipse with major and minor axes a and b, how does one find inscribed n-gon(s) of maximum perimeter?

Guess: the max perimeter inscribed n-gons for an ellipse are probably polygonal closed billiard paths (such polygons for n <=6 are shown on https://mathworld.wolfram.com/Billiards.html). If this guess is right, does it generalize to cases where a closed convex curve allows such polygonal billiard paths with m edges? The guess also implies a unique answer for an ellipse and n; that brings up the issue: given any point P on the ellipse boundary, to find the inscribed n-gon that necessarily contains P and has maximum perimeter.

2. When are the circumscribed n-gons of least perimeter and least area for an ellipse identical?

Note: one can also ask about inscribed and circumscribed n-gons in curves of constant width.

It appears that the max area inscribed n-gon for an ellipse is not unique - if one inscribes a regular n-gon in a circle with radius a (there are infinitely many orientations for this inscribed regular n-gon) and then compresses the circle along one direction into an ellipse with minor axis b, the inscribed regular n-gon gets pressed into an n-gon that seems to be a max area inscribed n-gon for the (a,b) ellipse.

1. Given an integer n and an ellipse with major and minor axes a and b, how does one find and characterize inscribed n-gon(s) of maximum perimeter?

Guess: the max perimeter inscribed n-gons for an ellipse are probably polygonal closed billiard paths (such polygons for n <=6 are shown on https://mathworld.wolfram.com/Billiards.html). If this guess is right, does it generalize to cases where a closed convex curve allows such polygonal billiard paths with m edges? The guess also implies a unique answer for an ellipse and n; that brings up the issue: given any point P on the ellipse boundary, to find the inscribed n-gon that necessarily contains P and has maximum perimeter.

2. When are the circumscribed n-gons of least perimeter and least area for an ellipse identical?

Note: one can also ask about inscribed and circumscribed n-gons in curves of constant width.

It appears that the max area inscribed n-gon for an ellipse is not unique - if one inscribes a regular n-gon in a circle with radius a (there are infinitely many orientations for this inscribed regular n-gon) and then compresses the circle along one direction into an ellipse with minor axis b, the inscribed regular n-gon gets pressed into an n-gon that seems to be a max area inscribed n-gon for the (a,b) ellipse.

1. Given an integer n and an ellipse with major and minor axes a and b, how does one find inscribed n-gon(s) of maximum perimeter?

Guess: the max perimeter inscribed n-gons for an ellipse are probably polygonal closed billiard paths (such polygons for n <=6 are shown on https://mathworld.wolfram.com/Billiards.html). If this guess is right, does it generalize to cases where a closed convex curve allows such polygonal billiard paths with m edges?

2. When are the circumscribed n-gons of least perimeter and least area for an ellipse identical?

Note: one can also ask about inscribed and circumscribed n-gons in curves of constant width.

It appears that the max area inscribed n-gon for an ellipse is not unique - if one inscribes a regular n-gon in a circle with radius a (there are infinitely many orientations for this inscribed regular n-gon) and then compresses the circle along one direction into an ellipse with minor axis b, the inscribed regular n-gon gets pressed into an n-gon that seems to be a max area inscribed n-gon for the (a,b) ellipse.

1. Given an integer n and an ellipse with major and minor axes a and b, how does one find inscribed n-gon(s) of maximum perimeter?

Guess: the max perimeter inscribed n-gons for an ellipse are probably polygonal closed billiard paths (such polygons for n <=6 are shown on https://mathworld.wolfram.com/Billiards.html). If this guess is right, does it generalize to cases where a closed convex curve allows such polygonal billiard paths with m edges? The guess also implies a unique answer for an ellipse and n; that brings up the issue: given any point P on the ellipse boundary, to find the inscribed n-gon that necessarily contains P and has maximum perimeter.

2. When are the circumscribed n-gons of least perimeter and least area for an ellipse identical?

Note: one can also ask about inscribed and circumscribed n-gons in curves of constant width.