Lines that cut 2-d object into equal parts, from Zach Star video

Question

In this interesting video from Zach Star https://www.youtube.com/watch?v=IJumRmwYsN4, he falsify the following:

False Claim every 2-d shape (open connected) has a point with the property that every line passing throught it, divide the shape in two part of equal area.

As the video shows the claim is false already for the equilateral triangle.

At this point some questions comes up naturally.

What are the shapes with this cutting point property? An obvious class is that of shape with a central symmetry, but does there exist others?
For every point in the plane there exist a line that pass throught it and cut the object in half, but for some points in the interior there exists multiple lines. Can we characterise this set of point?

You included good context, but try to also include some of your work/ what you find about the problem as seen in this MSE guide — Тyma Gaidash, Jul 17 '21 at 11:42

Acccumulation · Answer 1 · 2021-07-14T23:41:55.400

1

Let $d(\theta)$ be the difference in areas of the two sides of the line.

If you describe the shape in polar coordinates, with the boundary being defined by some function $r(\theta)$, then $d'(\theta) = \left|\frac{r(\theta)-r(\theta+\pi)}2\right|$. So the shape must have the property that $\forall \theta: r(\theta)=r(\theta +\pi)$. So the "cutting point" property is equivalent to being symmetric about the pole.

edited Jul 14 '21 at 23:41

answered Jul 13 '21 at 23:50

Acccumulation

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Very nice!! Is there a way to deal also with non smooth boundary? – Matteo Testa Jul 14 '21 at 00:31
Also maybe the correct formula is $d'(\theta)=\frac{r(\theta)-r(\theta+\pi)}{2}$, – Matteo Testa Jul 14 '21 at 08:16

Lines that cut 2-d object into equal parts, from Zach Star video

1 Answers1