We add a bit to On maximum perimeter triangles inscribed in convex regions with one vertex fixed. Let C be a convex planar region and P a point on its boundary.

Are there convex shapes C other than (any) ellipse such that: the area of the max area inscribed triangle with one vertex at P remains constant as P moves around boundary of C? If "yes", what could be said about them?

Are there convex shapes C other than the circle such that: the perimeter of the max perimeter inscribed triangle with one vertex at P remains constant as P moves around boundary of C?

Note: Motivation for these questions is the existence of infinitely many constant width closed curves (beginning with the Reuleaux triangle) other than the circle. And as was noted in above linked page, one can think of analogous questions on smallest triangles that *contain* C.

General remark: One could also revisit other properties of circles/ellipses and check if there are other families of convex regions which share those. A couple of questions are here: https://math.stackexchange.com/questions/4562022/on-sufficient-conditions-for-planar-convex-regions-to-be-circular-disks