Polynomial-gauge and polynomial-radial star bodies are dense, admit O(1/degree) approximation with optimal constants, and yield algorithms for slice and width computations.
Ragozin,Polynomial approximation on compact manifolds and homogeneous spaces, Trans
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Approximation of starshaped sets using polynomials
Polynomial-gauge and polynomial-radial star bodies are dense, admit O(1/degree) approximation with optimal constants, and yield algorithms for slice and width computations.