intermediate value theorem | If a continuous function times the value fa) and times f(b) assumes it must have any value s in between fa) and f(b) also in the interval [a, b] accept.
The same applies to the pizza and the salami spread on it. However, a real pizza is not always perfectly round. It may have indentations, especially if it has been shaped by hand. In this case, it is still possible to divide them fairly. The proof procedure is almost identical: you start with a cut that cuts the dough, cheese and tomato sauce in half. Then you let the knife (or the pizza roller) rotate – but no longer around a fixed point. If you turn the cutting line a bit, you may have to move the knife up, down, left or right a little to continue halving the dough. But the main argument remains the same: After the cutting line has been rotated by 180 degrees, you are back in the starting position with reversed salami portions for the left and right halves. You can use the intermediate value theorem again to prove the conjecture: Yes, every salami pizza can be divided fairly into two halves.
Sharing a ham sandwich
However, if you now put a basil leaf on the pizza, i.e. add a third component, the fair bisection is generally no longer possible – at least if you view the entire pizza as a flat plane. In two dimensions, you can exactly bisect exactly two objects by a straight cut. And so Steinhaus asked himself whether this can be transferred to three dimensions: Can one find a cutting plane to bisect three objects in three-dimensional space?
Unfortunately, in the three-dimensional case, the intermediate value theorem alone does not get us anywhere. Because for this one would have to define a starting plane to which one returns by rotating around an axis. In doing so, one would prove that at some point in the rotation, the objects were bisected. In three dimensions, however, there is no clear axis of rotation, but several, which is why the argument does not work without further ado. But one of Steinhaus’ protégés, Stefan Banach (1892–1945), found another way to prove the conjecture. For this he used the Borsuk-Ulam theorem, which I have already introduced in this column.
The Borsuk-Ulam theorem states, among other things, that there are always two diametrically opposite points on earth where the same temperature and air pressure prevail. Similar to halving a pizza, this theorem has to do with continuous functions (temperature and barometric pressure in this case) and geometry (the earth as a sphere). More formally, the Borsuk-Ulam theorem states: For any two-dimensional continuous function f(x,y) on a sphere there is a point (away) on its surface, for which applies f(a,b) = f(−a, −b). Banach realized that in the ham sandwich problem, you can also use a sphere to bisect the three components.