Splitting a convex set through its center

Three days ago I raised the question How unevenly can you split a convex set through its center? Scott had asked in the comments what kind of sets split most unevenly through their centers. I didn’t have an answer at the time, but now I have something.

Start with a triangle. Draw a line from the middle of the base up to the top. The center of mass is located 1/3 of the way up that line.

Now draw a horizontal line through the center of mass. The triangle above this line is similar to the entire triangle, but will have 2/3 of the original height. By similarity, it also has 2/3 of the base, and so it has 4/9 of the area of the entire triangle. The portion below the center of mass has 5/9 of the area.

Now let’s go up a dimension and imagine a tetrahedron sitting on a table. Draw a line from the center of the base up to the top. The center of mass for the tetrahedron will be located along that line, 1/4 of the way up. Pass a plane through the center of mass parallel to the table. This forms a new tetrahedron above the cutting plane, similar to the original tetrahedron. The new tetrahedron has 3/4 of the height, and by similarity, it has (3/4)³ = 27/64 of the original volume. Of course the solid below the cutting plane then has 37/64 of the original volume.

Now consider the general case. The center of mass of an n-simplex is located 1/(n+1) of the way up along a line running from the center of the base to the top. Cut the n-simplex through the center of mass with a hyperplane parallel to the base. The n-simplex above the hyperplane has n/(n+1) of the original height and (n/(n+1))ⁿ of the original volume. As n goes to infinity, this expression converges to 1/e.

In summary, we’ve shown that a convex set in Rⁿ can have at least 1 – (n/(n+1))ⁿ of its volume on one side of a hyperplane through the center of mass by constructing such a set, namely any n-simplex. And as n goes to infinity, this volume approaches 1 – 1/e. We have not shown that an n-simplex is the worst case in each dimension n, though I suspect this is true.  If you take a ball in Rⁿ, any plane through the center divides the ball exactly in half. In some sense, an n-simplex is as far as a convex set can get from a ball. It’s the least-round convex shape.

Update: Here’s a paper on the topic. B. Grunbaum, “Partitions of mass-distributions and convex bodies by hyperplanes,” Pacific Journal of Mathematics. 10, 1257–1261, 1960.