Here is an earlier discussion that could be related: On comparing planar convex regions of equal perimeter and area
We are broadly interested in placing two given planar convex regions so that the intersection has maximal area or perimeter. One could use translations, rotations and reflections(flipping over) of either region to achieve max intersection (a more constrained version of this post would disallow reflections).
Question 1: If $C_1$ and $C_2$ are planar convex regions both of unit area and some specified perimeter, how different can their maximum-area intersection and maximum-perimeter intersection be? The difference could be quantified by the ratio between the areas of the two optimal intersections, say. The two intersections might be most different for some specific value of the perimeter.
Note: Relaxing the constraint that the two regions have same perimeter could be of interest.
Question 2: Under what conditions are the maximum-area and maximum-perimeter intersections of two convex bodies identical? For example, does this claim hold?
Claim: "If two planar convex regions $C_1$ and $C_2$ have same area, perimeter and diameter, then the same arrangement of $C_1$ and $C_2$ maximizes the area and perimeter of their intersection."
The questions have obvious higher dimensional analogs.
Note added on March 17th, 2022: One can further ask under what conditions (if at all) will the centers of mass of $C_1$ and $C_2$ be necessarily coincident when the intersection of $C_1$ and $C_2$ is optimized? Indeed, for the case: maximizing the area of intersection of two general $C_1$ and $C_2$, the centers of mass need not coincide - for example, a thin rectangle of length a and an equilateral triangle of side a. But I don't know what happens once $C_1$ and $C_2$ have various properties -area, perimeter,...- equal.
And just like optimizing the intersections, one could consider optimizing the unions - eg. to minimize the area of the convex hull of the union of $C_1$ and $C_2$.
