Geodesic
Lower bounds for the number of hyperplanes separating two finite sets of points
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 2, pp. 210-222

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We consider the $NP$-hard polyhedral separability problem for two subsets $A$ and $B$ of points in general position in $\mathbb R^d$ with the fewest number of hyperplanes in the sense of boolean functions from a given class $\Sigma$. Both deterministic and probabilistic lower bounds are obtained for this number for two different classes of functions $\Sigma$.
Keywords: $k$-polyhedral separability, boolean function, monochromatic island, combinatorial discrepancy. 
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     author = {K. S. Kobylkin},
     title = {Lower bounds for the number of hyperplanes separating two finite sets of points},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {210--222},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2014_20_2_a16/}
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K. S. Kobylkin. Lower bounds for the number of hyperplanes separating two finite sets of points. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 2, pp. 210-222. http://geodesic.mathdoc.fr/item/TIMM_2014_20_2_a16/