We proceed from https://mathoverflow.net/questions/448712/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 https://mathoverflow.net/questions/448712/a-claim-on-the-concurrency-of-area-bisectors-of-planar-convex-regions, the situation with perimeter could be very different from area. 

Following https://mathoverflow.net/questions/437053/bisectors-and-partitioning-lines-for-convex-regions-defined-with-respect-to-the, 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).