Proof of Dudley's Convex Approximation

TL;DR

This paper offers a self-contained proof of Dudley's result that bounded convex bodies in Euclidean space can be approximated by a finite intersection of halfspaces, with bounds depending on the approximation accuracy and dimension.

Contribution

It provides a new, self-contained proof of Dudley's convex approximation theorem, clarifying the bounds and dependencies involved.

Findings

Convex bodies can be approximated by a finite intersection of halfspaces.

The number of halfspaces needed scales as _d(\u03b5^{-(d-1)/2}) with approximation accuracy.

The proof is self-contained, simplifying understanding of Dudley's original result.

Abstract

We provide a self contained proof of a result of Dudley [Dud64]} which shows that a bounded convex-body in $\Re^d$ can be $\varepsilon$-approximated, by the intersection of $O_d\bigl(\varepsilon^{-(d-1)/2} \bigr)$ halfspaces, where $O_d$ hides constants that depends on $d$.