Abstract
Inspired by Barany’s Colourful Caratheodory Theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.
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Deza, A., Huang, S., Stephen, T. et al. Colourful Simplicial Depth. Discrete Comput Geom 35, 597–615 (2006). https://doi.org/10.1007/s00454-006-1233-3
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DOI: https://doi.org/10.1007/s00454-006-1233-3