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I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines emanating from each vertex to points on the opposite edge so that each of the cells generated by these lines has diameter at most $\epsilon$. What is the minimum number of lines that I must use at each vertex?

More generally, I am partitioning the $N$-simplex (with coordinates $(p_1, p_2, \ldots, p_N)$, $p_i \ge 0$, and $\sum_i p_i = 1$) using hyperplanes in the set $\cup_{i,j=1}^N \{p_i = r_\ell p_j\}_{\ell=1}^L$ for some set $\{r_\ell \}_{\ell=1}^L$ that I can choose. A cell is a set of points in the simplex characterized by the intersection of half-spaces generated by all the hyperplanes. What is the minimum value of $L$ needed so that the diameter of each cell is less than $\epsilon$? Is it possible to characterize the choice of $r_i$?

I'd be very appreciative of any references you may have. Happy to give more specifics or more details of the context.

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  • $\begingroup$ Do you consider the diameter in the Euclidean metric induced from ambient space, where each edge has length $\sqrt 2$? And do you just want an upper bound on that minimum? Determining it on the nose might be tricky. $\endgroup$
    – Sebastian Goette
    Commented Nov 20, 2015 at 7:32
  • $\begingroup$ Yes, the diameter in the Euclidean metric in the ambient space is what I mean. I was asking about the exact upper bound (in case the answer to this problem is already known), but a good upper bound on the maximum would be useful. In two dimensions, I believe it's not too difficult to compute an upper bound assuming the $r_\ell$ are such that the lines emanating from each vertex intersect the opposite edge in equally spaced points. But, I imagine there is a better choice of $\{r_\ell\}$, and I am not sure how to go about analyzing that. $\endgroup$
    – User123321
    Commented Nov 20, 2015 at 16:26

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