Classifiers are often used to detect miscreant activities. We study how an adversary can systematically query a classifier to elicit information that allows the adversary to evade detection while incurring a near-minimal cost of modifying their intended malfeasance. We generalize the theory of Lowd and Meek (2005) to the family of convex-inducing classifiers that partition input space into two sets one of which is convex. We present query algorithms for this family that construct undetected instances of approximately minimal cost using only polynomially-many queries in the dimension of the space and in the level of approximation. Our results demonstrate that near-optimal evasion can be accomplished without reverse-engineering the classifier’s decision boundary. We also consider general lp costs and show that near-optimal evasion on the family of convex-inducing classifiers is generally efficient for both positive and negative convexity for all levels of approximation if p=1.