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

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

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

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 n continuously integrable functions:

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

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

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

(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^[3] 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.^[4] The Hobby–Rice theorem implies that this can be done with n cuts.