We proceed from A claim on the concurrency of area bisectors of planar convex regions

This question is somewhat broad.

Background: 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11 describes several results on lines that divide the area of a given convex planar region in the ratio $t : (1-t)$ where $0<t<1/2$ (area bisectors are those lines when $t=1/2$). In the same spirit, I have the following

Question. What is known about lines that divide the outer boundary (perimeter) of a planar convex region into 2 connected pieces in the ratio $t:1-t$ - the envelope of these dividing lines for a given $t$, whether these envelopes have cusps, self intersections and so forth?

Remarks: For convex regions with straight portions of the boundary, some of these dividing lines might lie flush with the boundary for small values of t but that need not rule out interesting results. In the light of the answer and comment (from Fedor Petrov) in A claim on the concurrency of area bisectors of planar convex regions, the situation with perimeter could be very different from area.

Following Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia, one can also ask about families of lines that break off a piece with moment of inertia a specified fraction of a given convex planar body (the MI being defined, say, about an axis perpendicular to the plane of the body about the center of mass of the piece).