We investigate the sample size needed to infer a separating line between two convex planar regions using Valiant's model of the complexity of learning from random examples . A theorem proved in  using the Vapnik-Chervonenkis dimension gives an O((1/ε)ln(1/ε)) upper bound on the sample size sufficient to infer a separating line with error less than ε between two convex planar regions. This theorem requires that with high probability any separating line consistent with such a sample have small error. The present paper gives a lower bound showing that under this requirement the sample size cannot be improved. It is further shown that if this requirement is weakened to require only that a particular line which is tangent to the convex hulls of the sample points in the two regions have small error then the ln(1/ε) term can be eliminated from the upper bound. © 1992.
Blumer, A., & Littlestone, N. (1989). Learning faster than promised by the Vapnik-Chervonenkis dimension. Discrete Applied Mathematics, 24(1–3), 47–53. https://doi.org/10.1016/0166-218X(92)90271-B