This earlier post attempted to quantify the difference between a pair of planar convex regions of equal area and perimeter using Hausdorff distance: On comparing planar convex regions of equal perimeter and area. Here we consider other ways.
Given 2 planar convex regions $C_1$ and $C_2$, one can think of placing them such that
- the area of their intersection is maximized or
- area of the convex hull of their union is minimized.
Question 1: Which pairs $\{C_1, C_2\}$ both of unit area and same specified perimeter give
- least area for their intersection when $C_1$ and $C_2$ are placed so that this intersection area is maximized
- maximum area of the convex hull of their union when $C_1$ and $C_2$ are positioned such that this hull is minimized?
Remarks: Basically we are asking how far apart a convex pair of unit area and equal perimeter can be as measured by their optimal intersection or union. One would guess the answer to (1) above are a pair of triangles with one of them isosceles. I don't know what happens if in addition to area and perimeter, $C_1$ and $C_2$ also should have equal diameter.
Note 1: Instead of using area to measure the difference between planar regions, one can consider other quantities such as perimeter and ask variants of above question. And there are natural analogs in higher dimensions.
Question 2: Is there a provably finite set of quantities {area, perimeter, diameter,..} such that for any $\{C_1,C_2\}$ pair with all of these quantities equal, the positioning of $C_1$ and $C_2$ that maximizes the area of their intersection automatically minimizes the area of the convex hull of their union?
