When deep neural networks became state of the art image classifiers, numerous max pooling operations were an important component of the architecture. However, modern computer vision networks typically have few, if any, max pooling operations. To understand whether this trend is justified, we develop a mathematical framework analyzing ReLU based approximations of max pooling, and prove a sense in which max pooling cannot be replicated. We formulate and analyze a novel class of optimal approximations, and find that the residual can be made exponentially small in the kernel size, but only with an exponentially wide approximation. This work gives a theoretical basis for understanding the reduced use of max pooling in newer architectures. It also enables us to establish an empirical observation about natural images: since max pooling does not seem necessary, the inputs on which max pooling is distinct – those with a large difference between the max and other values – are not prevalent.