The classical ham sandwich theorem says that given $n$ measurable sets in $\mathbb{R}^n$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single $(n − 1)$-dimensional hyperplane.
Does the theorem remain true if we replace measurable sets with measurable integrable real-valued functions on $\mathbb{R}^n$ and we wish for the hyperplane to divide the integrals of the functions in half? The classical theorem is the case where the functions are characteristic functions of sets in $\mathbb{R}^n$.
More precisely. let $f_1,\dots,f_n$ be real-valued functions in $L^1 (\mathbb{R}^n,m)$ (where $m$ is Lebesgue measure). Can we find an $(n − 1)$-dimensional hyperplane dividing $\mathbb{R}^n$ into two half-spaces $L_1,L_2$, such that $\int_{L_1} f_i dm= \int_{L_2} f_i dm$ for all $i=1,\dots,n$?
My feeling is that this is false, but I'm having trouble finding an elegant counterexample.