  We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a
random split tree of size $n$. We prove that, for $n$ large, the largest of
these $k$ Voronoi cells contains most of the vertices, while the sizes of the
remaining ones are essentially all of order $n\exp(-\mathrm{const}\sqrt{\log
n})$. This discrepancy persists if we modify the definition of the Voronoi
cells by (a) introducing random edge lengths (with suitable moment
assumptions), and (b) assigning different "influence" parameters (called
"speeds" in the paper) to each of the $k$ vertices. Our findings are in
contrast to corresponding results on random uniform trees and on the continuum
random tree, where it is known that the vector of the relative sizes of the $k$
Voronoi cells is asymptotically uniformly distributed on the
$(k-1)$-dimensional simplex.
