Optimal non-adaptive approximation of convex bodies by polytopes

TL;DR

This paper investigates the limits of non-adaptive algorithms for approximating convex bodies by polytopes, focusing on projection functions, and establishes their optimality in the context of approximation accuracy.

Contribution

It introduces and analyzes optimal non-adaptive algorithms based on projection functions for approximating convex bodies, filling a gap in existing methods.

Findings

Optimal non-adaptive algorithms using projection functions are identified.

These algorithms achieve the best possible approximation accuracy for arbitrary convex bodies.

The study clarifies the limitations of non-adaptive approaches compared to adaptive methods.

Abstract

In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the order algorithms produces polytopes for which the accuracy in Hausdorff metric is inversely proportional to the number of vertices (faces) in the degree of 2/(d-1). Numerical approximation algorithms can be adaptive (active) when the vertices or faces are constructed successively, depending on the information obtained in the process of approximation, and non-adaptive (passive) when parameters of algorithms are defined on the basis of a priory information available. Approximation algorithms differ in the use of operations applied to the approximated body. Most common are indicator, support and distance (Minkowski) functions calculations. Some optimal active algorithms for arbitrary bodies approximation are known using support or distance function calculation operation. Optimal passive algorithms for smooth bodies approximation are known using support function calculation operation and extremal curvature information. It is known that there are no optimal non-adaptive algorithms for arbitrary bodies approximation using support function calculation operation. We consider optimal non-adaptive algorithms for arbitrary bodies approximation using projection function calculation operation.