Hobby-Rice theorem

In mathematics, and in particular the study of necklace splitting problems, the Hobby-Rice theorem is a result that is useful in establishing the existence of certain solutions.

The theorem

If $g\_1,ldots,g\_k:\; [0,1]\; longrightarrowBbb\{R\}$ are given continuously integrable functions then there exist $0=z\_0leq\; z\_1leqldotsleq\; z\_kleq\; z\_\{k+1\}=1$ and $delta\_1,ldots,delta\_kinleft\{+1,-1\; ight\}$ such that $sum\_\{i=1\}^\{k\}delta\_iint\_\{z=z\_\{i-1^\{z\_i\}\; g\_j(z),dz=0$ for all $1leq\; jleq\; k$.

references

C. R. Hobby and J. R. Rice 1965. "A moment problem in $L\_1$ approximation. Proceedings of the American Mathematical Society, 16:665-670