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

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?

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.