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Proof of Dudley's Convex Approximation

Sariel Har-Peled, Mitchell Jones

TL;DR

This work provides a self-contained proof of Dudley’s convex-approximation result: a bounded convex body $C$ in $\mathbb{R}^d$ can be approximated within Hausdorff distance $\varepsilon$ by an intersection $D$ of $O_d(\varepsilon^{-(d-1)/2})$ halfspaces. The construction selects a maximal $\delta$-packing $Q$ on a radius-3 sphere with $\delta=\sqrt{\varepsilon}$, defines supporting halfspaces $h_q$ through the boundary projection $u_n$ orthogonal to $u-u_n$, and sets $D=\bigcap_{q\in Q} h_q$, ensuring $C\subseteq D\subseteq C_{\oplus\varepsilon}$ with $|Q|=\Theta_d(\varepsilon^{-(d-1)/2})$. A projection-contraction argument guarantees the reverse inclusion, with detailed geometric analysis of boundary points to bound their distance to $C$ by $\varepsilon$. This clarifies Dudley’s theorem in a self-contained manner and provides an explicit, dimension-dependent expression for the number of required halfspaces.

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 $d$.

Proof of Dudley's Convex Approximation

TL;DR

This work provides a self-contained proof of Dudley’s convex-approximation result: a bounded convex body in can be approximated within Hausdorff distance by an intersection of halfspaces. The construction selects a maximal -packing on a radius-3 sphere with , defines supporting halfspaces through the boundary projection orthogonal to , and sets , ensuring with . A projection-contraction argument guarantees the reverse inclusion, with detailed geometric analysis of boundary points to bound their distance to by . This clarifies Dudley’s theorem in a self-contained manner and provides an explicit, dimension-dependent expression for the number of required halfspaces.

Abstract

We provide a self contained proof of a result of Dudley [Dud64]} which shows that a bounded convex-body in can be -approximated, by the intersection of halfspaces, where hides constants that depends on .

Paper Structure

This paper contains 2 sections, 1 theorem, 4 equations, 6 figures.

Table of Contents

  1. Statement and proof
  2. More details

Key Result

Theorem 1.4

For $d \geq 2$, let $\mathsf{C}$ be a closed convex body in $\mathbb{R}^d$, such that $\mathsf{C}$ is contained in the ball of radius $1$ centered at the origin. For a parameter $\varepsilon \in (0,1)$, one can compute a convex body $\mathsf{D}$, which is the intersection of $O_d(1/\varepsilon^{(d-1

Figures (6)

  • Figure 1.1:
  • Figure 1.2: The triangle formed by $x,y$ and the origin.
  • Figure 1.3: The plane $f$.
  • Figure 1.4:
  • Figure 1.5: The angle $\gamma$ is minimized if $x$ and $y$ are as short as possible (i.e., $2$), and $\left\| {x} - {y} \right\|$ is maximized (i.e., $2\delta \leq 2$).
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1.1
  • Theorem 1.4: d-mescs-74
  • Claim 1.5
  • Claim 1.6