We introduce polystar bodies: compact starshaped sets whose gauge or radial functions are expressible by polynomials, enabling tractable computations, such as that of intersection bodies. We prove that polystar bodies are uniformly dense in starshaped sets and obtain asymptotically optimal approximation guarantees. We develop tools for the construction of polystar approximations and illustrate them via several computational examples, including numerical estimations of largest volume slices and widths.
Preprints
Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming
This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set. This slice-volume task is formulated as an infinite-dimensional linear program in continuous functions, inspired by prior work in volume computation of semialgebraic sets. A convergent sequence of upper-bounds to the maximal slice volume are computed using the moment-Sum-of-Squares hierarchy of semidefinite programs in increasing size. The computational complexity of this scheme can be reduced by utilizing topological structure (in dimensions 2, 3, 4, 8) and symmetry. This numerical convergence can be accelerated through the introduction of redundant Stokes-based constraints. Demonstrations of slice-volume calculation are performed on example sets.
@preprint{miller2024maximizing,title={{Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming}},author={Miller, Jared and Meroni, Chiara and Tacchi, Matteo and Velasco, Mauricio},year={2024},tag={slice},arxiv={2403.04438},archiveprefix={arXiv},primaryclass={math.OC},url={https://arxiv.org/abs/2403.04438},html={https://arxiv.org/abs/2403.04438},bibtex_show={true},code={https://github.com/jarmill/slice_volume},slides={Slice_Volume_Presentation__MoPAT_Konstanz_.pdf}}