Hobby–Rice 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).

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.