We propose a family of quasi-linear discriminants that outperform current large-margin methods in sliding window visual object detection and open set recognition tasks. In these tasks the classification problems are both numerically imbalanced-positive (object class) training and test windows are much rarer than negative (non-class) ones and geometrically asymmetric-the positive samples typically form compact, visually-coherent groups while negatives are much more diverse, including anything at all that is not a well-centred sample from the target class. It is difficult to cover such negative classes using training samples, and doubly so in 'open set' applications where run-time negatives may stem from classes that were not seen at all during training. So there is a need for discriminants whose decision regions focus on tightly circumscribing the positive class, while still taking account of negatives in zones where the two classes overlap. This paper introduces a family of quasi-linear "polyhedral conic" discriminants whose positive regions are distorted L1 balls. The methods have properties and run-time complexities comparable to linear Support Vector Machines (SVMs), and they can be trained from either binary or positive-only samples using constrained quadratic programs related to SVMs. Our experiments show that they significantly outperform both linear SVMs and existing one-class discriminants on a wide range of object detection, open set recognition and conventional closed-set classification tasks.