## Electronic Communications in Probability

### A note on the richness of convex hulls of VC classes

#### Abstract

We prove the existence of a class $A$ of subsets of $\mathbb{R}^d$ of VC dimension 1 such that the symmetric convex hull $F$ of the class of characteristic functions of sets in $A$ is rich in the following sense. For any absolutely continuous probability measure $\mu$ on $\mathbb{R}^d$, measurable set $B$ and $\varepsilon \gt 0$, there exists a function $f$ in $F$ such that the measure of the symmetric difference of $B$ and the set where $f$ is positive is less than $\varepsilon$. The question was motivated by the investigation of the theoretical properties of certain algorithms in machine learning.

#### Article information

Source
Electron. Commun. Probab., Volume 8 (2003), paper no. 18, 167-169.

Dates
Accepted: 17 December 2003
First available in Project Euclid: 18 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1463608902

Digital Object Identifier
doi:10.1214/ECP.v8-1097

Mathematical Reviews number (MathSciNet)
MR2042755

Zentralblatt MATH identifier
1095.28006

Rights

#### Citation

Lugosi, Gábor; Mendelson, Shahar; Koltchinskii, Vladimir. A note on the richness of convex hulls of VC classes. Electron. Commun. Probab. 8 (2003), paper no. 18, 167--169. doi:10.1214/ECP.v8-1097. https://projecteuclid.org/euclid.ecp/1463608902

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