Colourful Simplical Depth
The simplicial depth of a point p relative to a finite
set S in Rd
is the number of simplices from S that contain p.
This quantity is
studied in statistics since the point of maximum depth is a d-dimensional
median. By considering a colourful generalization of simplicial depth
we improve Barany's lower bound for the depth of the median of n points
in Rd, and encounter some nice geometric and
algorithmic questions.