Hobby–Rice theorem

Short description: Necklace splitting problem

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; a simplified proof was given in 1976 by A. Pinkus.

The theorem

Define a partition of the interval [0,1] as a division of the interval into [math]\displaystyle{ n+1 }[/math] subintervals by as an increasing sequence of [math]\displaystyle{ n }[/math] numbers:

[math]\displaystyle{ 0=z_0 \lt \underbrace{z_1 \lt \dotsb \lt z_n} \lt z_{n+1} = 1 }[/math]

Define a signed partition as a partition in which each subinterval [math]\displaystyle{ i }[/math] has an associated sign [math]\displaystyle{ \delta_i }[/math]:

[math]\displaystyle{ \delta_1,\dotsc,\delta_{k+1}\in\left\{+1,-1\right\} }[/math]

The Hobby–Rice theorem says that for every n continuously integrable functions:

[math]\displaystyle{ g_1,\dotsc,g_n\colon[0,1]\longrightarrow\mathbb{R} }[/math]

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

[math]\displaystyle{ \sum_{i=1}^{n+1}\delta_i\!\int_{z_{i-1}}^{z_i} g_j(z)\,dz=0\text{ for }1\leq j\leq n. }[/math]

(in other words: for each of the n 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 in 1987.

Suppose the interval [0,1] is a cake. There are n partners and each of the n functions is a value-density function of one partner. We want to divide the cake into 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. The Hobby–Rice theorem implies that this can be done with n cuts.