Abstract: We introduce the idea of strict minimizers for geometric distortion measures used in shape interpolation, deformation, parametrization, and other applications involving geometric mappings. The L∞-norm ensures the tightest possible control on the worst-case distortion. Unfortunately, it does not yield a unique solution and does not distinguish between solutions with high or low distortion below the maximum. The strict minimizer is a minimal L∞-norm solution, which always prioritizes higher distortion reduction. We propose practical algorithms for computing strict minimizers. We also offer an efficient algorithm for L∞ optimization based on the ARAP energy. This algorithm can be used on its own or as a building block for an ARAP strict minimizer. We demonstrate that these algorithms lead to significant improvements in quality.