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.