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It has been known that given two probability distributions $\mu_1$ and $\mu_2$ (let us say, they are smooth for simplicity), there is a hyperplane that divides the domain into two regions (denoted as $E_1$, $E_2$) so that these two probability distributions have the same integration on these two regions, i.e., $\int_{E_j} \mu_k(x) dx = 1/2$ for any $j, k = 1, 2$. This is basically the ham sandwich theorem.

My question is that (let us consider 2D for simplicity):

Suppose we iteratively find ham sandwich cuts (we allow e.g., curves in 2D), e.g., we apply the same process again for $E_1$ and $E_2$ separately and divide the domain into four parts, eight parts ... on which $\mu_1$ and $\mu_2$ have the same integration. Can these curves/lines form streamlines of an ODE? An related vague question is that under what conditions can we ensure continuity and differentiability of these ham sandwich cuts after we iteratively find them?

I am not in the field of geometry nor topology. Any references or thoughts are appreciated and would be helpful. Thank you.

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  • $\begingroup$ If you just use straight lines to make your cuts, they are the solutions of an ODE: free particles in space. Do you want to pick the ODE first, and then make the cuts? Can you say more maybe about what sort of ODE you want? If the measures both lie entirely along a streamline, and the streamlines foliate (1st order ODE), you won't be able to do this $\endgroup$
    – Ben McKay
    Commented Sep 10, 2021 at 17:07
  • $\begingroup$ I want to find those cuts first (i.e., apply ham sandwich theorem, or some other results...), and then prove that these cuts/curves are streamlines of an ODE (i.e., prove the existence of an ODE $\dot{X} = f(X)$ with $f$ being continuous such that its streamlines are exactly the cuts we find by iteratively applying ham sandwich theorem). My intuition is that I hope those cuts do not interact in a strange way (e.g., totally horizontal and vertical cuts intersecting with each other in a strange way) by imposing regularity assumption on probability measures, but I don't know where to get started. $\endgroup$
    – user483904
    Commented Sep 10, 2021 at 18:40
  • $\begingroup$ The ODE that I need is that $\dot{X} = f(X)$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$ is Lipschitz continuous. But I am totally fine with piecewise Lipschitz continuous. $\endgroup$
    – user483904
    Commented Sep 10, 2021 at 18:46

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