## The hamburger theorem.(English)Zbl 1380.05068

Summary: We generalize the ham sandwich theorem to $$d + 1$$ measures on $$\mathbb R^d$$ as follows. Let $$\mu_1$$, $$\mu_2$$,$$\dots$$,$$\mu_{d+1}$$ be absolutely continuous finite Borel measures on $$\mathbb R^d$$. Let $$\omega_i = \mu_i(\mathbb R^d)$$ for $$i \in [d + 1]$$, $$\omega =\min\{\omega_i;i \in [d + 1]\}$$ and assume that $$\sum^{d+1}_{j=1} \omega_j = 1$$. Assume that $$\omega_i \leq 1/d$$ for every $$i \in [d + 1]$$. Then there exists a hyperplane $$h$$ such that each open halfspace $$H$$ defined by $$h$$ satisfies $$\mu_i(H) \leq (\sum^{d+1}_{j=1} \mu_j(H))/d$$ for every $$i \in [d + 1]$$ and $$\sum^{d+1}_{j=1} \mu_j(H) \geq\min\{1/2, 1 - d\omega\} \geq 1/(d + 1)$$. As a consequence we obtain that every $$(d + 1)$$-colored set of $$nd$$ points in $$\mathbb R^d$$ such that no color is used for more than $$n$$ points can be partitioned into $$n$$ disjoint rainbow $$(d - 1)$$-dimensional simplices.

### MSC:

 05C15 Coloring of graphs and hypergraphs
Full Text:

### References:

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