The walk of a knife

Question

"A knife is slowly moved parallel to itself over the top of a cake. At each instant the knife is poised so that it could cut a unique slice of the cake. As time goes by the potential slice increases monotonely from nothing until it becomes the entire cake" (Dubins & Spanier, "How to cut a cake fairly", 1961).

Define $C$ as the entire cake and define $K_C(t)$ as the subset of $C$ that has already been covered by the knife at time $t$. This function has the following properties:

1. It is a monotonically increasing function of $t$.
2. $K_C(0)=\phi$ and $K_C(T)=C$ (where $T$ is a finite positive number).
3. The Lebesgue measure $L(K_C(t))$ is a continuous increasing function of $t$.
4. For every non-atomic measure $M$ on $C$, the function $M(K_C(t))$ is a continuous non-decreasing function of $t$.

My questions are:

• Is there a formal term for a function $K_C(t)$ having all these properties?
• Does such a function exist for every "cake" (set) $C$? If not, under what conditions it is possible to define a function $K_C(t)$ over a set given $C$?
• Is property 4 required in the definition of $K_C(t)$, or can it be inferred from the previous properties? (I ask this question because I would like to have a simpler definition of the "knife walk", without using "for every non-atomic measure").

Answer 1

The usual setting for this procedure is that $C$ is a set with positive, but finite, Lebesgue measure. Now $4$ holds for measures absolutely continuous with respect to Lebesgue measures but not with respect to every nonatomic measure.

Suppose you move the knife in $\mathbb{R}^2$ with the blade parallel to the vertical axis. The measure that is uniformly distributed over the line segment $\{0\}\times [0,1]$ will lead to a jump in measure once the knife has reached the horizontal axis.

The implicit formalization of a moving knife is that you take a linear functional $l$ on some $\mathbb{R}^n$ and look at the measure $\mu$ of the intersection of $C$ with the halfspaces $H_t=\{x\in\mathbb{R}^n:l(x)\leq t\}$. So $K_C(t):\mathbb{R}\to\mathbb{R}$ is given by

$$K_t(C)=\mu(C\cap H_t).$$

The counterexample given is essentially Example $1$ of

Hill, Theodore P., and Kent E. Morrison. "Cutting cakes carefully." The College Mathematics Journal 41.4 (2010): 281-288.

The paper has an interesting history, that might tell you more about the field than you might want to know.