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