Approximation of starshaped sets using polynomials

2025/05/30 by Chiara Meroni, Meroni, Chiara, Jared Miller +3
Mathematics · #51M16 #52-08 #52A40 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2505.24352

openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound
• A note on Bourgain's slicing problem
• Harmonic Hierarchies for Polynomial Optimization
• Intersection Bodies of Polytopes
• Convergence of SDP hierarchies for polynomial optimization on the hypersphere
• Computing Fourier Transforms and Convolutions on the 2-Sphere

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