Detecting the pupil center plays a key role in human-computer interaction, especially for gaze tracking. The conventional deep learning-based method for this problem is to train a convolutional neural network (CNN), which takes the eye image as the input and gives the pupil center as a regression result. In this paper, we propose an indirect use of the CNN for the task, which first segments the pupil region by a CNN as a classification problem, and then finds the center of the segmented region. This is based on the observation that CNN works more robustly for the pupil segmentation than for the pupil center-point regression when the inputs are noisy IR images. Specifically, we use the UNet model for the segmentation of pupil regions in IR images and then find the pupil center as the center of mass of the segment. In designing the loss function for the segmentation, we propose a new loss term that encodes the convex shape-prior for enhancing the robustness to noise. Precisely, we penalize not only the deviation of each predicted pixel from the ground truth label but also the non-convex shape of pupils caused by the noise and reflection. For the training, we make a new dataset of 111,581 images with hand-labeled pupil regions from 29 IR eye video sequences. We also label commonly used datasets (ExCuSe and ElSe dataset) that are considered real-world noisy ones to validate our method. Experiments show that the proposed method performs better than the conventional methods that directly find the pupil center as a regression result. INDEX TERMS Convex shape prior, deep learning, pupil segmentation, U-Net.