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 ${\displaystyle k+1}$ subintervals:

${\displaystyle 0=z_{0}<z_{1}<\dotsb <z_{k}<z_{k+1}=1}$

Define a signed partition as a partition in which each subinterval ${\displaystyle i}$ has an associated sign ${\displaystyle \delta _{i}}$:

${\displaystyle \delta _{1},\dotsc ,\delta _{k+1}\in \left\{+1,-1\right\}}$

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

${\displaystyle g_{1},\dotsc ,g_{k}\colon [0,1]\longrightarrow \mathbb {R} }$

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

${\displaystyle \sum _{i=1}^{k+1}\delta _{i}\!\int _{z_{i-1}}^{z_{i}}g_{j}(z)\,dz=0{\text{ for }}1\leq j\leq k.}$

(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. This fair-division challenge is sometimes referred to as the consensus-halving problem.[4] 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.
4. F.W. Simmons and F.E. Su (2003). "Consensus-halving via theorems of Borsuk-Ulam and Tucker" (PDF). Mathematical Social Sciences. 45: 15–25. doi:10.1016/S0165-4896(02)00087-2.