Given a planar convex region $C$ (could be smooth or polygonal), an area bisector of $C$ is any line that partitions $C$ into 2 pieces of equal area. A 'fair bisector' is a line that partitions $C$ into 2 pieces of equal area and equal perimeter.
Thru every point on the boundary of $C$, an area bisector can be drawn (for a description of their properties, please see 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11). But it can be seen that a convex planar region can have just a single fair bisector (eg. for a thin isosceles triangle, the only fair bisector is the bisector of its apex angle) or a finite number of them (in which case, their number is necessarily odd as can be seen from simple continuity arguments; see reference at the top) or infinitely many.
Observations: For regions with a center of symmetry such as a circular disk or ellipse or regular polygon with even number of sides, all fair bisectors are concurrent. But, numerically, we see that for a general convex region $C$ with finitely many fair bisectors, the fair bisectors are not necessarily concurrent but usually very close to being so. Clearly,for a general $C$ with exactly 3 fair bisectors, they determine a small triangular region deep in the interior of $C$. For $C$s with more fair bisectors, their many possible intersections will divide the interior of $C$ into many regions. Let us refer to the union of those regions which do not share the outer boundary of $C$ as the 'core' of $C$. The core must lie deep inside $C$.
For which convex shape of $C$ is the area of the 'core' of $C$ the largest as a fraction of the area of $C$? Intuitively, a relatively large core is a measure of the asymmetry of $C$. Can one say (say) that such a shape is always one with exactly 3 fair bisectors?
Generalizing a bit, what about lines that break off the same fraction $t$ of the area and outer boundary length of $C$? For a circular disk, it appears that only for $t=1/2$, we have such lines (any diameter). Are there $C$'s for which such lines exist for several (maybe even arbitrarily many) different values of $t$?
Guess: All centrally symmetric convex regions (rectangles, ellipses,...) appear to give such only one single partitioning line that divides both area and outer perimeter in same ration - only for $t=1/2$. But general convex regions with no symmetry might give infinitely many such lines - one such partitioning line for each orientation - and a different value $t$ for each orientation. And the set of these lines might even have interesting envelopes.
These questions have obvious higher dimensional analogs.