Ref: A center of convex planar regions based on chords
The above discussion quotes the definition of 'centralness coefficient' and defines a center of a planar convex region. 1/2 is the least possible value of centralness coefficient among all convex planar regions. This value holds for all triangles. In this post, we consider another quantity to measure how 'rounded' a convex region is and partitions of convex regions based on this quantity. Note that a thin rectangle has centralness (it is centrally symmetric) but is not very rounded.
Question: For every point P in the interior of a planar convex region C, consider the ratio: distance from P to closest point on the boundary / distance from P to the farthest boundary point. Let us call this the depth ratio of P. Consider the point(s) in C that maximize the depth ratio ('depth center', say)and note the value of this ratio there. How does one partition a planar convex region into n convex pieces such that every piece has the same maximum depth ratio?
Remarks: For any C, the depth center(s) appear to lie on the medial axis. Even when C is a triangle, we don't have a unique depth center but the incenter seems to be a solution. Unlike the centralness coefficient, the maximum depth ratio of a triangle region goes to 0 as the triangle tends to become degenerate.
Further Question: Are there any other properties that are shared by the pieces in a partition that gives pieces with same maximum depth ratio? And continuing from On cutting convex regions with average values of quantities minimized, one can ask if maximizing the minimum of the max depth ratio among pieces necessarily makes it equal across pieces.
