# On largest convex m-gons contained in a given convex n-gon where m < n

This post is the inside-out variant of On smallest convex m-gons that contain a given n-gon where m<n

1. Given a convex n-gon region P, and an m less than n, how to find the max area convex m-gon Q contained within P?

2. Same question as above with 'perimeter' replacing 'area'.

A natural approach for both questions would be to select m from among the n vertices of P and to connect them with edges into a convex polygon - and repeat for all such m-subsets. Not sure how close to optimal the answer would be (the method itself looks inefficient).

As special cases, one can try to characterize and find the least area/perimeter triangles and convex quadrilaterals that are contained within P.

The maximum area inscribed convex $$m$$-gon can always be found on the vertices. To do this, just consider moving one vertex at a time; we gain area the further we move this point from the diagonal connecting the two vertices adjacent to it. Maximizing distance from this diagonal involves finding the extremum of a linear function within the convex $$n$$-gon, which is always attained at a vertex.
I suspect you can make a similar argument for the perimeter case, by thinking about the equi-perimeter ellipses and where those would first meet the $$n$$-gon.