The high energy physics unfolding problem is an important statistical inverse problem in data analysis at the Large Hadron Collider (LHC) at CERN. The goal of unfolding is to make nonparametric inferences about a particle spectrum from measurements smeared by the finite resolution of the particle detectors. Previous unfolding methods use ad hoc discretization and regularization, resulting in confidence intervals that can have significantly lower coverage than expected. Instead of regularizing using a roughness penalty or early stopping, we impose physically justified shape constraints: positivity, monotonicity, and convexity. We quantify the uncertainty by constructing a nonparametric confidence set for the true spectrum consisting of all those spectra that satisfy the shape constraints and fit the smeared data within a given confidence level. Solving the resulting semi-infinite optimization problems yields confidence intervals that have guaranteed frequentist finite-sample cb overage in the important and challenging class of unfolding problems for steeply falling particle spectra. We demonstrate that the method is effective using simulations that mimic unfolding the inclusive jet transverse momentum spectrum at the LHC. The shape-constrained intervals provide usefully tight conservative confidence intervals, while the conventional methods suffer from severe undercoverage. Joint work with Philip B. Stark (UC Berkeley).