One-sided epsilon-approximants

TL;DR

The paper introduces the concept of one-sided epsilon-approximants for finite point sets in Euclidean space, demonstrating their existence with size bounds depending on epsilon and dimension, contrasting with traditional two-sided approximants.

Contribution

It establishes the existence of size-bounded one-sided weak epsilon-approximants for any finite point set in Euclidean space, a novel result differing from classical two-sided approximants.

Findings

Existence of one-sided epsilon-approximants with size bounds

Contrast with two-sided epsilon-approximants

Applicable to all finite point sets in Euclidean space

Abstract

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show that, in contrast with the usual (two-sided) weak $\varepsilon$-approximants, for every set $P\subset \mathbb{R}^d$ there exists a one-sided weak $\varepsilon$-approximant of size bounded by a function of $\varepsilon$ and $d$.