Approximation of starshaped sets using polynomials

TL;DR

This paper introduces polystar bodies, a class of starshaped sets with polynomial gauge functions, providing a new approach for approximating and computing properties of starshaped sets with proven density and approximation guarantees.

Contribution

The paper defines polystar bodies, proves their density in starshaped sets, and develops computational tools for their approximation, enabling efficient geometric computations.

Findings

Polystar bodies are dense in starshaped sets.

Asymptotically optimal approximation guarantees are established.

Numerical examples demonstrate practical computation of geometric properties.

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.