Here’s a surprising theorem. Suppose you have a convex set in Rn you pass a plane through its center of mass. How much of the volume of the set can be on one side? No more than about 63%. (Precisely, 1 – 1/e.) This holds for any dimension n and for any direction through the center. I don’t have a reference for this theorem except that it is mentioned near the end of lecture 5 in this course.
Update: See Splitting a convex set through its center for an illustration and a partial proof.