Draw a line through the centroid of a triangle. It cuts the triangle into two pieces. The classical Winternitz theorem says the area ratio of the two pieces is bounded: the smaller piece is at least 4/9 of the total area, achieved by equilateral triangles. But what about the perimeter ratio?
Berele and Catoiu (arXiv: 2603.23653) answer: the perimeter fraction of the smaller piece ranges over the interval (3/10, 4/9]. The upper bound 4/9 is achieved — equilateral triangles with lines parallel to a side split the perimeter exactly 4:5. The lower bound 3/10 is approached but never reached, by degenerate triangles approaching the ratio 5:4:1.
The gap between 3/10 and 4/9 is the space of all possible worst-case perimeter splits across all triangles. Equilateral triangles are the most balanced — any centroid line splits their perimeter nearly evenly. The 5-4-1 triangle is the most imbalanced — a centroid line can create pieces whose perimeters differ by a factor approaching 7:3.
Area and perimeter behave differently because area is a bulk quantity (scales with the square of linear dimensions) while perimeter is a boundary quantity (scales linearly). A line through the centroid necessarily balances area well because the centroid is the center of mass. But perimeter doesn't concentrate at the centroid — long thin triangles have perimeter distributed far from the centroid, allowing extreme splits.
The through-claim: the center of mass balances weight but not boundary. The centroid is defined by area, and area splits are constrained by that definition. But perimeter is a different measure on the same shape, and the centroid has no special relationship to it. The same point that guarantees a 4/9 area split can allow a 3/10 perimeter split. What is central for one measure can be peripheral for another.
Berele & Catoiu, 2603.23653. Metric geometry / Winternitz theorem / triangle geometry / centroid / perimeter bounds.