Lines segmenting convex planar regions - some questions

A pair of claims on area bisectors and perimeter bisectors of convex planar regions were posted at mathoverflow here and a further question is here .

Let us define a width of a planar convex region C as the distance between two parallel lines that just touch C. A width bisector is aline that is parallel to a pair of parallel lines that are tangent to C and is at same distance from both these lines - basically, it divides a width into two equal parts.

Claim: All width bisectors of a convex planar C are concurrent if an only if C is centrally symmetric.

Question: What can be said about families of lines that divide all widths in ratio t:1-t where t is between 0 and 1/2?

Guess: If we are looking at families of lines that segment area(equally, perimeter or widths) in some specified ratio, for no convex planar region can families of these dividing lines be concurrent if the specified ratio is other than 1:1.

Just like the envelopes of families of lines that segment area - these have been discussed by Fuchs and Tabachnikov - one could ask for properties of the envelopes of the other families of segmenting lines too.
-----------
The ratio between areas (or perimeters, least widths or diameters) of these envelopes (or the convex hulls of these envelopes) and those of C could serve a measure of how 'uncentered' C is. Yaglom and Boltyanski have talked about 'centralness' based on chords and proved that it has the least value for triangles. It seems likely that the envelopes of the segmenting lines with reference to the different quantities mentioned above (area etc) would be quite different for the same C. The centralness measures based on these ratios too could have intresting implications.
----------
Higher dimensional equivalents of these questions are obvious.