Hobby–Rice theorem

In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles R. Hobby and John R. Rice;^[1] a simplified proof was given in 1976 by A. Pinkus.^[2]

The theorem

Given an integer k, define a partition of the interval [0,1] as a sequence of numbers which divide the interval to $k+1$ subintervals:

$0=z_0 < z_1 < \dotsb < z_k < z_{k+1} = 1$

Define a signed partition as a partition in which each subinterval $i$ has an associated sign $\delta_i$:

$\delta_1,\dotsc,\delta_{k+1}\in\left\{+1,-1\right\}$

The Hobby-Rice theorem says that for every k continuously integrable functions:

$g_1,\dotsc,g_k\colon[0,1]\longrightarrow\Bbb{R}$

there exists a signed partition of [0,1] such that:

$\sum_{i=1}^{k+1}\delta_i\!\int_{z_{i-1}}^{z_i} g_j(z)\,dz=0\text{ for }1\leq j\leq k.$

(in other words: for each of the k functions, its integral over the positive subintervals equals its integral over the negative subintervals).

Application to fair division

The theorem was used by Noga Alon in the context of necklace splitting^[3] in 1987.

Suppose the interval [0,1] is a cake. There are k partners and each of the k functions is a value-density function of one partner. We want to divide the cake to two parts such that all partners agree that the parts have the same value. The Hobby-Rice theorem implies that this can be done with k cuts.

References

1. Hobby, C. R.; Rice, J. R. (1965). "A moment problem in L₁ approximation". Proceedings of the American Mathematical Society (American Mathematical Society) 16 (4): 665–670. doi:10.2307/2033900. JSTOR 2033900.
2. Pinkus, Allan (1976). "A simple proof of the Hobby-Rice theorem". Proceedings of the American Mathematical Society (American Mathematical Society) 60 (1): 82–84. doi:10.2307/2041117. JSTOR 2041117.
3. Alon, Noga (1987). "Splitting Necklaces". Advances in Mathematics 63 (3): 247–253. doi:10.1016/0001-8708(87)90055-7.