This paper presents a new approach to the problem of detecting out-of-distribution (OOD) inputs in image classifications with deep convolutional networks. We leverage so-called boundary samples to enforce low confidence (maximum softmax probabilities) for inputs far away from the training data. In particular, we propose the boundary optimised samples (named BoS) training algorithm for generating them. Unlike existing approaches, it does not require extra generative adversarial network, but achieves the goal by simply back propagating the gradient of an appropriately designed loss function to the input samples. At the end of the BoS training, all the boundary samples are in principle located on a specific level hypersurface with respect to the designed loss. Our contributions are i) the BoS training as an efficient alternative to generate boundary samples, ii) a robust algorithm therewith to enforce low confidence for OOD samples, and iii) experiments demonstrating improved OOD detection over the baseline. We show the performance using standard datasets for training and different test sets including Fashion MNIST, EMNIST, SVHN, and CIFAR-100, preceded by evaluations with a synthetic 2-dimensional dataset that provide an insight for the new procedure.