I am studying a problem of finding a bound on the diameter of cells in a partitioned simplex (with points $(p_1, \dots, p_n)$, $p_i \ge 0$, and $\sum_i p_i = 1$). The partitions are formed by hyperplanes characterized by $p_i = q_k p_j$ for some predetermined set $\{q_k\}_{k=1}^K$ with $0\leq q_k \leq \infty$ and $i,j\in\{1, \dots, n\}$.  (For instance, in the 2-simplex, these are lines emanating from a vertex to the opposite side.)  

Suppose that the $\{q_k\}$ are such that the hyperplanes are "evenly spaced."  By evenly spaced, I mean that $\{q_k\}$ is such that $q_k / (1+q_k) = k / K$.  (In the 2-simplex, this corresponds to the lines dividing the opposite edge into equally spaced segments.)

What is the maximal diameter of one of the cells of this partition, as a function of $K$ and $n$?  (An answer even for the 2-simplex would be very helpful!)  Suppose for concreteness that we are looking at the Euclidean metric induced from the ambient space, with all edges having side length $\sqrt{2}$.