The exploratory and interactive nature of modern data analysis often introduces selection bias and poses challenges to traditional statistical inference methods. To address selection bias, a common approach is to condition on the selection event. However, this often results in a conditional distribution that is intractable and requires Markov chain Monte Carlo (MCMC) sampling for inference. Notably, some of the most widely used selection algorithms yield selection events that can be characterized as polyhedra, such as the lasso for variable selection and the $\varepsilon$-greedy algorithm for multi-armed bandit problems. This talk will present a method that is tailored for conducting inference following polyhedral selection. The proposed method transforms the variables constrained within a polyhedron into variables within a unit cube, allowing for exact sampling. Compared to MCMC, this method offers superior speed and accuracy. Furthermore, it facilitates the computation of maximum likelihood estimators based on selection-adjusted likelihoods. Numerical results demonstrate the enhanced performance of the proposed method compared to alternative approaches for selective inference.