# Unbalance in cake-cutting

In cake-cutting, we cut a round cake into $$m$$ pieces by $$n$$ cuts. If we seek maximum $$m$$ pieces with $$n$$ cuts, we may observe that the area unbalance increases greatly with $$m$$. Could we define a mathematical relationship among the unbalance, pieces and cuts?

Let's describe it precisely in math notions. Given a unit disk with total area $$\pi$$, for a configuration $$c$$ of $$n$$ straight lines, $$c$$ cuts the disk into $$m$$ pieces, and the area of each piece is $$A_i$$.

Define the portion of each piece by $$p_i = \frac{A_i}{\pi}$$

The area unbalance may be described by entropy: $$S_2 = - \sum_{i=1}^{m}p_i \ln p_i$$

The arc length of the outer pieces is also unbalanced, a similar entropy $$S_1$$ can be introduced:

$$S_1 = - \sum_jq_j \ln q_j$$

where $$q_j = \frac{L_j}{2 \pi}$$ is the portion of the segments of the circumference.

Could we describe the relationship among $$S_1$$, $$S_2$$, $$m$$ and $$n$$? And given $$m$$ and $$n$$, which cut $$c$$ leads to the least unbalance $$S_1$$ and $$S_2$$?

$$n$$ lines divide the plane into at most $$m = \frac{n(n+1)}{2}+1$$ pieces, which is known as the lazy caterer's sequence. This results in $$\frac{n(n-1)}{2}$$ intersections of two lines each, or equivalently $$n-1$$ intersections on every line. Notice that for odd $$n$$ the regular $$\{n/\frac{n-1}{2}\}$$ star polygons satisfy these criteria:
$$n$$ lines have $$2n$$ degrees of freedom which can be represented by the $$x$$ and $$y$$ coordinates of each point of the star. Now, depending on $$r := \sqrt{x^2+y^2} < 1$$ we can easily compute $$A_i$$ and $$L_j$$ because of the rotational symmetry and minimize the entropies.